Ne ≈ 20. Marker subsets were also tested.Genomic selection (GS) may improve selection response over conventional pedigree-based selection if markers capture more detailed information than pedigrees in recently domesticated tree species and/or make it more cost effective. Genomic prediction accuracies using 1748 trees and 6932 SNPs representative of as many distinct gene loci were determined for growth and wood traits in white spruce, within and between environments and breeding groups (BG), each with an effective size of r = 0.71–0.79) and moderately high for growth (r = 0.52–0.69) traits, in line with trends in heritabilities. For both classes of traits, these accuracies achieved between 83% and 92% of those obtained with phenotypes and pedigree information. Prediction into untested environments remained moderately high for wood (r ≥ 0.61) but dropped significantly for growth (r ≥ 0.24) traits, emphasizing the need to phenotype in all test environments and model genotype-by-environment interactions for growth traits. Removing relatedness between CV sets sharply decreased prediction accuracies for all traits and subpopulations, falling near zero between BGs with no known shared ancestry. For marker subsets, similar patterns were observed but with lower prediction accuracies.Model fits and/or cross-validation (CV) prediction accuracies for ridge regression (RR) and the least absolute shrinkage and selection operator models approached those of pedigree-based models. With strong relatedness between CV sets, prediction accuracies for RR within environment and BG were high for wood (Given the need for high relatedness between CV sets to obtain good prediction accuracies, we recommend to build GS models for prediction within the same breeding population only. Breeding groups could be merged to build genomic prediction models as long as the total effective population size does not exceed 50 individuals in order to obtain high prediction accuracy such as that obtained in the present study. A number of markers limited to a few hundred would not negatively impact prediction accuracies, but these could decrease more rapidly over generations. The most promising short-term approach for genomic selection would likely be the selection of superior individuals within large full-sib families vegetatively propagated to implement multiclonal forestry. Genomic selection GS, has beenTheoretical–6 and emNe), as it is easily controlled by the plant and tree breeders = G = C ⊗ A and var[T] = R = E ⊗ I for the additive genetic and residual effects, respectively, where ⊗ is the direct (kronecker) product and C and E are matrices containing the additive genetic and residual (co)variance parameters to be estimated as:where the total vectors of random effects have covariance structures var[(A) but that the residual covariance across sites is zero because measures took place on different individuals in different environments.which indicates that there is a genetic covariance across sites (with information flowing through 8) variance. Weak, proper priors drawn from the inverse-Wishart distribution were set to the phenotypic variance divided by the number of random effects in the model with the degree of belief parameter set to 1 and 2 for uni- and bivariate models, respectively. Mixing and stationarity of the chains was assessed with trace plots of samples from the posterior distribution using the coda package[h2) and genetic correlations between traits (r12) were calculated asEBVs; âi) obtained from the bivariate analyses were stored on the site where phenotypes were observed and were subsequently used to determine prediction accuracies with univariate SNP-based GEBVs (see next).Model parameters were sampled from the posterior distribution using the MCMCglmm v2.16 package in R v2. package. PosteriFollowing bivariate analyses, phenotypes were adjusted for fixed block effects and scaled by within-site phenotypic standard deviation in order to account for  differences in phenotypic variation for traits on different sites. No differences in genetic means between the breeding groups were detected. Phenotypes were then combined across sites for use in marker regressions, assuming that traits on different sites were a single trait with a genetic correlation equal to unity.In multiple-marker regression, many SNPs are simultaneously estimated as random effects in the individual tree model:ys is a vector of adjusted and standardized phenotypic measurements at age 17, 1 is a vector of ones, μ is an intercept, ai is the vector of additive genetic (breeding) values, uk is a vector of random marker effects with the n x m incidence matrix containing marker covariates coded as Zki =  so that the sum of marker effects approximates the individual (additive) genomic estimated breeding value (GEBV)ei is the vector of residual error effects. Additive genetic (polygenic) and error effects were assumed to be distributed as in eq. χ2 prior density with degrees of freedom (df) and scale (S) parameters set towhere  = 0, 1,  so that ended in.τ2, which is in turn controlled by the prior distribution of the regularization parameter (p(λ))[−4 yielding λ = 57.5 for our data. Large values of λ produce more informative (sharper) priors on marker effects[Two types of marker regression were performed that differ in the prior distribution of SNP effects. In ridge regression (RR), all SNP effects are assumed to have a common variance by assigning a Gaussian prior asr (p(λ)). The regr (p(λ)), with pa effects, which w effects in R v2. effects.A) or genomic relationship (K) matrix using the EMMAX algorithm[F-test is constructed for each SNP to test the hypothesis that the SNP effect uk ≠ 0.Single-marker regression  was also conducted to test the hypothesis that individual marker loci were in partial or complete linkage disequilibrium with a causal gene and to delineate subsets of markers for further marker regressions. Marker effects were treated as fixed effects (unlike in eq. lgorithm in the rlgorithm in R (v2lgorithm. VariancP < 0.05) using SMR were retained, controlling for relatedness with A or K. Next, the 600 SNPs with largest absolute effects were retained from the full model  using RR and pedigree information (FM-RRA), which approximated the maximum number of significant SNPs from SMR. SNP effects were estimated in a subpopulation (see below) and subsequently re-evaluated using RR in the same (e.g. BG1) or complementary subpopulation (e.g. BG1 → BG2). In the former case, SNP effects treated as fixed effects in SMR may be biased when estimated and validated in the same non-independent data[Traits were modeled using 1) all individuals with all SNPs and/or the pedigree and 2) subpopulations of individuals with all or subsets of SNPs and/or the pedigree in cross-validation (CV). SNP subsets for CV were retained for reevaluation using RR in five different ways. First, the SNPs with the largest absolute effects ) between the sum of CV SNP estimates (ĝ) or GEBVs and the additive genetic (breeding) value (â) or EBVs estimated using the bivariate, pedigree-based model and all phenotypic data, which was considered the best estimate of the “true” breeding value.For cross-validation (CV) analyses, individuals (CV1) or full-sib families (CV2) were assigned to 10 folds randomly. In each of 10 rounds of CV, 9 folds acted as the training set in which SNP effects were estimated in order to predict the genetic values of individuals in the remaining fold, whose phenotypes had been withheld. For models tested across subpopulations (BG1 → BG2 or S1 → S2), that is training the models in one subpopulation and testing them in the other one, each CV round used 9 of 10 folds as the training set from subpopulation 1, with a different fold used as the testing set for each round of CV from subpopulation 2. Prediction accuracies were calculated as the correlation  – within-breeding group CV simulates a situation where related individuals have already been phenotyped on both (or similar) sites, as BG members were distributed across sites. GEI could lower prediction accuracy if present.BG1 → BG2 – between-breeding group CV removes all known relatedness between CV sets, establishes baseline accuracy due to (short-range) LD and thus tests if the model is transferable beyond the study material. As members of both BGs are present in both environments, GEI could have an impact.S1 (or S2) – within-site CV simulates a situation where models are trained by pooling individuals across BGs, with no known relatedness between BG. Evaluation occurs within site, thus GEI cannot affect the results.S1 → S2 – between-site CV simulates a situation where no phenotypes have been observed on related individuals in the second environment, testing the assumption that the traits on different sites have a genetic correlation of one.r12) was not equal to 1 across sites (Table r12 = 0.73), to high . This implied that, biologically, the GEI was not strong enough to warrant the more complex (bivariate) model for wood traits (less so for growth traits) and that data might be pooled across sites and considered as a single trait.Univariate heritabilities (not shown) were similar to their bivariate counterparts Table , suggestK matrix revealed that < 5% of the variation was captured by the first eigenvector. This indicated that population structure was not important among the breeding groups and that A or K was sufficient to capture the relatedness among individuals. The lack of phenotypic differences observed between the two breeding groups, BG1 and BG2, supported this lack of structure. The K matrix captured a low level of relatedness between the two BGs  values for half- and full-sibs of 0.20  and 0.44 , respectively.Spectral decomposition of the n = 1,748) and SNPs  simultaneously with ridge regression (FM-RR), which assumes a common variance across SNPs, only improved model fit (lower DIC) and reduced the error variance for ADEN, compared with the pedigree-based model (FM-A)  were very high  and, given the limited improvement in LASSO fits, this model was not further considered.Allowing marker-specific shrinkage with the LASSO (FM-L and FM-LA) resulted in only very minor improvement in model fit and reduced error variance over the corresponding models using a common SNP variance (FM-RR and FM-RRA) for all traits Table . The varo Table . Again, For CV1, prediction accuracies were high for pedigree-only (CV1-A) models because the CV sets were highly related. Marker-only models (CV1-RR) captured a high proportion (0.87 ≤ 0.92) of the CV1-A prediction accuracy for all traits, when modeled within breeding groups (BG1) or environment (S1) Table . ModelinOverall in cross-validation using family assignment (CV2) where relatedness between CV sets did not exceed those of half-sibs, prediction accuracies decreased sharply (e.g. 0.12 – 0.41 for BG1 and S1 with RR) for all models, traits and subpopulations Table , compareRAN) or highest minor allele frequency (CVSSHMAF) subsets for growth traits, but differences for wood traits were less clear. Prediction accuracies were slightly higher when SNP effects had been treated as fixed effects (CVSSA or CVSSK) and then re-evaluated in the same population (BG1 or S1). This trend suggests that, in some cases, completely eliminating many markers of supposedly zero effect did not hamper much the estimation of the fewer “true” marker effects, as was previously hypothesized from results obtained using the LASSO models. However, in many instances, the accuracy estimates were slightly lower than those obtained with the full set of markers. This might be due to the removal of some markers that were in LD with QTLs associated with the traits of interest.For SNP subsets that were identified and re-evaluated within the same population (BG1 or S1), all model-based methods used to select subsets performed relatively well Table . SNP subr > 0.44), but low for growth traits  with no known relatedness to BG1, prediction accuracies fell to near zero for all subsets and traits  and distinct loci  used in this study and a white spruce genome size of ~ 2,100 cM[Ne = 30 – 60 and a training set with n = 1,000 individuals, one should theoretically achieve GS model accuracies of 0.55 – 0.70[e ≈ 20) or for the same environment when combining both breeding groups (Ne ≈ 40) using slightly smaller (mean n = 787) training sets. However, as for pedigree-based models, prediction accuracy of marker-based models decreased and fell near zero when close  or realized genomic relationships (CVSSK), or markers with the largest absolute effects (CVSSRRA)  or site (S1). Compared with the prediction accuracies obtained with all markers, a slight increase was indeed seen when using markers found significantly associated with traits when using either pedigree information (CVSS) Table . This inIn a recent study on wood and growth traits in loblolly pine full-sib families, no increase in prediction accuracy was reported for various subsets of markers selected after association studies or randomly selected markers up to a maximum number of about 3,400 markers. HoweverThe observation that GS prediction accuracies sharply decreased when training and validation individuals were completely unrelated (BG1 → BG2) may be due to various causes. The first could be related to the fact that the long-range LD (co-segregating linkage blocks) generated through controlled crosses among the closely related individuals making up the training set may not be present in a non-related testing set. Linkage studies have shown that marker effects are generally limited to the family (genetic background) in which they were estimated because m = 600 versus m = 6,932), but only slightly  might be needed in order to demonstrate the superiority of marker- over pedigree-based prediction accuracies[A complementary explanation for the loss of prediction accuracy among unrelated individuals could be that markers were largely capturing existing relationships between CV training and testing sets, 16. It curacies or that curacies. The lataij = 0.4 − 0.6[aij = 0.5 for full-sibs, and can be used to approximate the level of LD in a population[One way to think about the level of LD in a population is to consider the relationship matrix estimated with the markers, which can be partitioned into 1) the expected (known) relationships obtained from pedigree information, 2) the deviations around these relationship values due to Mendelian sampling, i.e. those due to the fact that each offspring receives a different set of genes from its parents and that each offspring is different from its expected degree of kinship, and 3) the sampling error. At high.4 − 0.6, 54 relapulation. Thus, tpulation, implyinpulation, which epulation. In thispulation. In our r12 = 0.83), which was reflected in high (BG1) and moderately high (S1 → S2) prediction accuracies when phenotypes obtained in the second site (S2) were included or not in the training model, respectively (Table The results obtained with the two CV scenarios that tested the impact of GEI (BG1 and S1 → S2) were different for wood and growth traits. Wood traits were relatively unaffected by GEI in the bivariate model  and low (S1 → S2) levels, when phenotypes measured in the second site (S2) were included or not in the training model, respectively Table . Such GEBorrowing correlated information  to make predictions across environments was found useful in wheat, which mThe bivariate model, which simultaneously considers the same trait measured on trees tested on two sites as different correlated traits, could be extended to a multivariate or reduced rank factor analytic model, 60 if tIn practice, our results indicate that future prospects for genomic selection in spruces should take place within populations of small effective size combining several breeding groups, and thus implicating individuals of high relatedness within breeding groups. To recapitulate, this is first because marker-based model accuracies do not hold when removing relatedness between CV sets, indicating that with the current marker densities the overall short-range LD between markers and causal loci is either too low to be of practical use or not sufficiently well captured. With the proposed approach, reduction in marker coverage as shown in Table The obvious caveat for the proposed approach is that it is not possible to clearly separate the contributions of both relatedness and long-range LD (co-segregation) from prediction accuracy. Long-range LD may still be important if its contribution to prediction accuracy does not decay as fast as that due to relatedness, especially with recurrent selection, i.e. in a population with the same genetic background. The use of multigenerational data might be one way to disentangle the two sources of information.A simple form of GS that does not depend on LD would consist of replacing the pedigree-based numerator relationship matrix with the genomic relationship matrix in the standard individual  BLUP model, or to bn = 200 and more), phenotype a fraction of the individuals (e.g. 25%), and select both phenotyped and unphenotyped individuals[Another simple GS scenario that does not depend on our ability to separate the source of prediction accuracy would be to conduct genomic prediction within full-sib families. In thisividuals for propividuals).Table 7In the short term, genomic selection would be most useful to select superior individuals within large full-sib families. These individuals would be reproduced by vegetative propagation techniques such as somatic embryogenesis (SE), or a mix of SE and rooted cuttings, in order to develop a multivarietal forestry program. With regard to genomic selection for recurrent selection at current marker densities in spruce populations, its efficiency would depend on the presence of close relatives in order to exploit the capture of relatedness and supposedly long-range LD. Thus, a simple marker-based approach that aim to uncover relatedness when pedigrees have not been recorded or are only partially known would seem to be the most cost-effective application of this technology. For such a scenario, the total number of SNPs required would be only a few hundreds to a few thousands. In the longer term, superior marker-over pedigree-based predictions may be achievable in descendant generations of closed populations as short-range LD increases and pedigrees provide less information. However, to achieve this goal, denser marker arrays, more individuals per family, and more sophisticated experiment designs and genomic selection models than tho


Although it has been suggested that rare coding variants could explain the substantial missing heritability, very few sequencing studies have been performed in rheumatoid arthritis (RA). We aimed to identify novel functional variants with rare to low frequency using targeted exon sequencing of RA in Korea.n = 1,217) and controls (n = 717). We conducted a single-marker association test and a gene-based analysis of rare variants. For meta-analysis or enrichment tests, we also used ethnically matched independent samples of Korean genome-wide association studies (GWAS)  or immunochip data .We analyzed targeted exon sequencing data of 398 genes selected from a multifaceted approach in Korean RA patients (VSTM1 showed the greatest association with RA (P = 7.80 × 10−4). In the enrichment test using Korean GWAS, although the significant signal appeared to be driven by total genic variants, we found no evidence for enriched association of coding variants only with RA.After stringent quality control, we analyzed 10,588 variants of 398 genes from 1,934 Korean RA case controls. We identified 13 nonsynonymous variants with nominal association in single-variant association tests. In a meta-analysis, we did not find any novel variant with genome-wide significance for RA risk. Using a gene-based approach, we identified 17 genes with nominal burden signals. Among them, We were unable to identify rare coding variants with large effect to explain the missing heritability for RA in the current targeted resequencing study. Our study raises skepticism about exon sequencing of targeted genes for complex diseases like RA.The online version of this article (doi:10.1186/s13075-014-0447-7) contains supplementary material, which is available to authorized users. Rheumatoid arthritis RA MIM 180300)) is a complex autoimmune disorder that results from both genetic and environmental risk factors [80300) is80300) isHLA-B, HLA-DRB1, and HLA-DPB1 affecting five amino acid positions [Nearly 60 RA risk loci were identified in several large studies including genome-wide association studies (GWAS) –6 and imositions ,10. The ositions . The othositions .et al. [It has been suggested that rare or low-frequency variants could explain the substantial unexplained heritability of many complex diseases, most of which were not fully captured using the previous conventional genotyping technology. Recently, Stahl et al.  inferredet al. [New genomic technologies, including next-generation sequencing (NGS), can provide a new approach for identification of rare variants. With advances in NGS technology, the role of rare or low-frequency variants in many complex diseases like RA can be investigated to better characterize the genetic architecture of the disease. Recently, several sequencing studies that have investigated common autoimmune diseases have shown that rare variants within genes containing common variants are associated with complex diseases –16. For et al.  performeHere, we aimed to identify novel functional variants with rare to low frequency using targeted exon sequencing in Korean RA, which dealt with hundreds of selected genes that were related to RA in various aspects such as previous identified genes from GWASs and immunochip data, literature reviews, and related pathways.A total of 1,252 RA cases were enrolled from the BAE cohort of Hanyang University Hospital for Rheumatic Diseases and satisfied the American College of Rheumatology 1987 classification criteria  for RA. We selected candidate genes using a comprehensive approach that included previous genetic and biological research, pathway databases, text-mining analysis, and animal-model databases  and performed high-throughput paired-end sequencing using a HiSeq2000 . The sequencing reads were mapped to the human reference genome, where the reference sequence was UCSC assembly hg19 (NCBI build 37.1) using Burrows-Wheeler Aligner (BWA) software . We thenP <0.01 in controls, and also eliminated samples that had insufficient call rates (<90%). We finally analyzed the 10,588 exonic variants of 1,934 samples for further single-variant association test, gene-based test, and pathway-based association enrichment test . We obtained an initial dataset with 50,247 variants from 666 targeted genes. We filtered the original genotype matrix by single nucleotide polymorphism (SNP) quality and depth coverage. The filtered genotype data contains genotype calls satisfying with practical guidance in rare variants analysis of complex trait association studies [P values were calculated using PLINK v1.07 software [To analyze for single-marker association with RA in targeted exon sequencing data, odds ratios (OR) and software  with adjP value of Hardy-Weinberg equilibrium (HWE) <10−4 and call rate >95% in cases and controls).For a meta-analysis  using a Korean RA GWAS  and iCHIP <0.05 in a single-variant test of sequencing data), we imputed common and low-frequency variants (MAF >0.5%) from the Korean GWAS data  by ShapeIt and IMPUTE2 with the 1000 Genomes Phase I reference panel. We performed logistic regressions for 1,000-times permuted phenotypes with the top 10 PCs as covariates by PLINK. Then, the numbers of genic, exonic, nonsynonymous, or synonymous variants reaching the Pthreshold <0.05 between observed and permuted data were compared by using a Fisher’s exact test.For an enrichment test for exonic SNPs in 77 newly identified genes , we performed both nonburden testing )  and burdP values for each SNV by logistic regression with adjustment for the top 10 PCs. To eliminate linkage disequilibrium (LD)-derived enrichment bias, we clumped the set of SNVs (r2 <0.4) in order of statistical significance. Then, we compared the number of SNV with P <Pthreshold between permuted datasets and empirical datasets by Fisher’s exact tests. Genes in each functional pathway were obtained from Ontology and KEGG.In pathway-based enrichment test of NGS data, we generated 1,000 permuted phenotype sets and their disease association Ethics approval was granted by the institutional ethics committee of Hanyang University in the Republic of Korea.Patient and control consents were obtained.After stringent quality control on the 666 targeted genes , and 6,605 SNVs were novel which were not identified in the 1000 Genome Project dataset  single-variant association test, (2) gene-based test for rare variants of which MAF are less than 0.05, and (3) pathway-based association enrichment test , with an SNV (rs1088680) within PRKCH showing the strongest association (P = 3.16 × 10−5)  exists or not, we performed an enrichment analysis for RA associations in all risk genes excluding 12 known RA risk loci using an independent dataset from our GWAS. Association of 41,454 SNVs within the 77 genes in the imputed Korean GWAS data was compared with the results after 1,000 case-control permutations by Fisher’s exact tests .Next, in an attempt to investigate whether an aggregate effect of 89 risk genes identified in NGS  = 1.04 × 10−32), we found no evidence for enriched association at coding variants only , nonsynonymous , and synonymous ) , most of which had two or more nonsynonymous rare variants, although they did not reach the threshold for significance after Bonferroni correction (P <1.2 × 10−4) ). The following analysis using only these validated samples showed that a set of the validated seven nonsynonymous variants of VSTM1 conferred a protective role in RA . The four variants of VSTM1 were primarily within the immunoglobulin-like domain among the coding regions. The two variants, A33T at the domain and D122N closed to the domain, were thought to be deleterious variants by PolyPhen2, suggesting that these may have a functionally protective role in RA  using nonburden tests ) , burden ion P <1. × 10−4 (Penrichment (nonsynonymous) = 8.55 × 10-4 and Penrichment (synonymous) = 0.166 at Pthreshold = 0.05) (data not shown) for the 335 genes. However, in the analysis for each pathway, we did not identify any pathway in which a significant enrichment for coding variants for RA existed at Pthreshold = 0.05.In the third analysis, we conducted a pathway enrichment analysis of coding variants (nonsynonymous and synonymous) within 335 novel genes using Ontology and KEGG, which were obtained after excluding the known RA risk loci (+/-250 kb) from the initial 398 genes in NGS. In this analysis of both common and rare variants, we observed weak but significant evidence of overall enrichment of coding SNVs  showed a negligible impact of rare autoimmune-locus coding variants on unexplained heritability (<3%) .A possible reason for this negative finding may be the limited number of targeted genes that were sequenced, which were 398 genes in the current study. Rare coding variants of the remaining genes across whole genome could participate in the missing genetic contribution to RA. The other potential reason may be genetic heterogeneity for RA, in which each individual or small subset of RA may have their particular rare causal variants. The current study revealed a lot of examples in which a specific rare variant was observed in only one or two individuals among 1,934 subjects. To discover these lots of ‘private’ causal variants that are susceptible only in small subsets of RA patients, deep sequencing of whole exome from very large RA cases, approximately thousands of individuals, would be required.VSTM1, a top signal gene driven by a gene-based test, is a glycoprotein primarily expressed in immune tissues, which can promote the differentiation and activation of Th17 cells [VSTM1  found in the current study were deleterious variants by PolyPhen2. Therefore, it is certainly worth replicating these variants or performing deep sequencing of the entire VSTM1 gene from an independent larger population, especially in Caucasians.17 cells . In addiVSTM1 and TaqMan genotyping of 37 variants from 200 samples. Finally, we included approximately 2,000 subjects for sequencing, which might not be enough of a sample size to discover rare variants. Consequently, this lack of power may lead most of the analysis in the study, such as the single-variant association analysis, gene-based tests, and enrichment tests, to be less significant with weak association.There are several limitations of this study. First, we performed a targeted exon sequencing study, which tends to generate more biased data than whole-exome or whole-genome sequencing studies. Additionally, we did not attempt to validate all rare variants identified by alternative methods, but rather performed it only for selected variants such as Sanger sequencing of rare nonsynonymous variants of However, to our knowledge, this study represents the largest sequencing study that evaluated the largest number of candidate genes with the largest case controls for RA until now. Despite negative findings, further replication of the possible single variants or rare variants in the study will be of some interest.We were unable to identify rare coding variants with large effect of 398 targeted genes. Despite much anticipation regarding missing heritability, our study raises skepticism about next-generation sequencing of targeted genes in order to discover rare variants with large effect for complex traits like RA. With the advance of genetic technology such as capturing, sequencing of targeted genes, and whole-exome or genome sequencing with a lot of subjects could define more details about the genetic architecture of RA in future.


The number of co-infections of a pathogen (multiplicity of infection or MOI) is a relevant parameter in epidemiology as it relates to transmission intensity. Notably, such quantities can be built into a metric in the context of disease control and prevention. Having applications to malaria in mind, we develop here a maximum-likelihood (ML) framework to estimate the quantities of interest at low computational and no additional costs to study designs or data collection. We show how the ML estimate for the quantities of interest and corresponding confidence-regions are obtained from multiple genetic loci. Assuming specifically that infections are rare and independent events, the number of infections per host follows a conditional Poisson distribution. Under this assumption, we show that a unique ML estimate for the parameter ( Infections are ubiquitous and ecologically complex processes. Indeed the chain of events conducing to the colonization and replication of parasites within a host involves many environmental, physiological, and genetic factors both in the host and the infectious agent. A common observation in many host-parasite interactions is that there are multiple genetically distinct lineages of the pathogen infecting the same individual host The observation of multiple genetic variants or multiplicity of infection (MOI) is indicative of the transmission dynamics since it allows for the co-transmission of different parasite variants or the overlap of several genetic variants due to multiple infectious contacts. Thus, the incidence of MOI or superparasitism per se is an important metric of exposure During the last two decades, the outcomes of intra-host dynamics have been the subject of several theoretical and experimental investigations exploring a broad spectrum of scenarios. Usually, such studies focus on major effects that different interconnected factors have in terms of parasite dispersion (parasite fitness) and/or the elicited manifestations of disease that may lead to an effect on the host's fitness Although it is possible to control or measure the number of distinctive parasite lineages in models and experimental settings e.g., a totalPlasmodium, Trypanosoma, Toxoplasma, and SchistosomaGiven the broad spectrum of genetic architectures observed in parasitic organisms, it is not possible to define a universal framework of MOI. E.g. HIV accumulates mutations at a rate that allows for the use of phylogenetic base methods More specifically, we further develop the maximum-likelihood framework introduced by The philosophy behind the method section's structure is the following. We first establish the general methods and then refine them assuming that the number of co-infections follows a conditional Poisson distribution. This structure embraces a better understanding of how to derive particular results for alternative choices to the Poisson distribution. Moreover, rigorous mathematical proofs are shifted to the appendix. Readers less interested in these technical details should feel free to skip them.We adapt the maximum-likelihood method of Assume Because we identify a pathogen with the allele at the considered locus, we will use the terms ‘lineage’ and ‘allele’ synonymously. In vector notation, the lineages' relative frequencies are In practice, Let Assume a total of It follows, that for a given allele-frequency distribution Of note, this conditions on the fact that each host is infected by at least one pathogen. The mean value of this distribution is Assuming the CPD (2), Consider a total of Since the  likelihood for observing these samples is Assuming the CPD for the number of lineages infecting a host, it is shown in Analysis (subsection 4.2) that the log-likelihood becomes Assumption 1Assume that the sum over the alleles' prevalences is larger than one, but not all alleles are  prevalent. In other words, more than one lineage is found in at least one infection, i.e.,  and not all lineages are found in every infection, i.e.,  for at least one .In the following We shall start by deriving the maximum likelihood (ML) estimates for the parameters of interest. Before we do so, we shall start by a rather intuitive observation.Not surprisingly Remark 1If at least one sample contains more than one allele, i.e.,is not the maximum likelihood estimate.To obtain the ML estimate for In the case of the conditional Poisson distribution (2) the entries of the Hessian matrix are derived in Analysis (subsection 4.4).Clearly, instead of (6) also  cf. e.g..Of note, in general, an ML estimate does neither necessarily exist, nor is it unique, not to mention that closed formulas typically do not exist. Unfortunately, assuming the CPD (2), the ML estimate indeed cannot be calculated explicitly. However, the estimate exists and is unique. Furthermore, although it can be straightforwardly derived by the above methods, the complexity of whole procedure can be greatly simplified.Result 1 Assume the conditional Poisson distribution (2) for . Under Assumption 1 there is a unique maximum likelihood estimate . The first component  is the unique positive solution of the equation.It is found by iteratingwhich converges monotonically and at quadratic rate from any initial valueThe maximum likelihood estimates of the allele frequencies are given by The result is proven in Analysis (subsection 5.1).For the sake of completeness we shall also consider the instances in which Assumption 1 is violated. In the first situation, only one pathogen lineage is found in each infection, i.e., there is no indication whatsoever of co-infections. The results are summarized in the following remark which is proven in Analysis (subsection 5.1).Remark 2Assume that each sample contains only one allele, i.e., . Then the ML estimates are  and .In the other non-generic case that all alleles are found in every sample an ML estimate does not exist, more precisely, it is Remark 3Assume  for all . Then the ML estimate is “” for every allelic distribution.A proof can be found in Analysis (subsection 5.1).Of note, the maximum likelihood has an intuitive interpretation. We summarize this as the following result which is proven in Analysis (subsection 5.1).Remark 4The maximum likelihood estimate  is the set of parameters for which the observed number of samples containing allele  equals its expectations, i.e., Hence, the maximum likelihood maximizes the expectation of the log-likelihood.Let We are interested in finding a confidence interval (CI) for Clearly, where , the confidence interval for  (based on the profile likelihood) is uniquely defined.The bounds of the confidence interval ( and) forare obtained by iteratingwhereandThere are exactly two possible solutions  and . The algorithm is converging quadratically for any initial values  sufficiently close to the one of the solutions.The proof is found in Analysis (subsection 5.2).Formally, the above result holds true in the non-generic cases As an alternative to the profile likelihood, one can use the asymptotic normality of the maximum likelihood to construct confidence intervals. Asymptotically the difference of the maximum likelihood  the asymptotic distribution of the parameters of interest Since Here, Of course, when using the expected Fisher information, With some algebraic manipulation it is possible to simplify the expressions for the confidence intervals assuming the CPD (2).Result 3Suppose the number of co-infections follow the conditional Poisson distribution (2) and that Assumption 1 holds. Then an asymptotic -confidence interval for  is given by Alternatively, the following formula, requires just the ML estimate for For a proof, see Analysis (subsection 5.3).In the non-generic case In the case Explicit formulas for the CIs of the allele frequencies are obtained similarly.Result 4Under the same assumptions as Result 3, an asymptotic -confidence interval for  is given by The proof can again be found in Analysis (subsection 5.3).In practice, data from several loci is typically available, each of which yields a different ML estimate or there might be some prior estimate for the parameters of interest. Depending on particular properties of the marker loci  different marker loci will lead to different ML estimates. Hence, it is desirable to test whether different estimates are significantly different. The confidence intervals can be adapted to test the parameters.vs. the alternative Clearly, at different marker loci, different alleles will segregate and the allele-frequency spectra will be very different. Hence, for the present purpose, it is meaningless to compare the allele frequencies at different loci. However, the estimate for The first test is rather straightforward. Since In other words, we reject the null hypothesis for any The corresponding p-value is given by To calculate the p-value, Result 5Suppose Assumption 1 and  holds. In the case of the conditional poisson distribution, the p-value under the null hypothesisis given by (24), where  is given by (4) with  and  given by where  is the solution of (16e) withThe solution  is found by iterating The proof is presented in Analysis (subsection 5.4).In case of Therefore, in the case of In the following, for any parameter choice vs. the alternative A test for the null hypothesis Note that it is legitimate to write Alternatively, the expected Fisher information Result 6Consider the score test for the null hypothesis  vs. the alternative  under the assumptions of Result 5. The test statistic based on the observed Fisher information is and that based on the expected Fisher information isThe p-values are  in either case. The frequencies  are derived as specified in Result 5.Of note, instead of (30) the ML estimate can be used as a plug-in estimate for the asymptotic variance, i.e., In summary one obtains:Remark 5Under the assumptions of Result 6, a test statistic for the null hypothesis  vs. the alternative  is whereandare sample size and number of alleles, in the data yielding the estimateThe proof is analogously to the one of Result 6.The test cannot be applied in the special cases A third test for the null hypothesis This implies Now, we shall consider again the CPD. An explicit expression for Result 7Under the assumptions of Result 5, the Wald test for the null hypothesis  vs. the alternative  has the test statistic based on the (expected or observed) Fisher information.The p-values are  in either case. Here,  and the frequencies  are derived as specified in Result 1.Alternatively, if the profile-likelihood estimate based on In the first case, using (53) implies that the test statistic changes to In the second case, (54) implies that the test statistic changes to Also the Wald test cannot be applied in the special cases Although - as we have seen - most of the theory works quite general, assuming a CPD for the number of co-infections permits to derive explicit results or, at least, reduces the complexity significantly. However, assuming a CPD might not be justified. Therefore, it is desirable to have a test for the model's fit. Namely, let Result 8To test : “the conditional Poisson distribution is justified” vs. : “the conditional Poisson distribution is not justified”, the test-statistic  can be used. The p-value is given by .It should be mentioned that the above test might perform poorly if the number of lineages or alleles P. falciparum-infected blood samples from endemic areas with different levels of malaria incidence. For each blood sample, parasite DNA was extracted and several microsatellite markers assayed.As an illustration, the methods are applied to three previously-described data sets mdr1, crt, dhfr, dhps in P. falciparum that are associated with selection for drug resistance) will have skewed allele-frequency distribution. Hence, using these markers might lead to artifacts and severe misinferences. In practice, a marker located on a chromosome not carrying a strongly selected gene (e.g. resistance-conferring gene), can be regarded to be neutral. Moreover, clinical samples from groups that will be compared need to consider confounding effects such as differences in treatment polices, control interventions, and changing transmission intensities . By not considering such effects, the estimates of MOI would be inappropriate. For these reasons, we only used parts of the available data sets.It is important to note beforehand that only (selectively) neutral markers should be included in the analysis. Namely, loci linked to others that are targets of selection  were included, resulting in 42 blood samples.The first data set emerged from a longitudinal study conducted in Asembo Bay, a hyper-endemic region in Kenya, and was described in The second data set described in dhps and dhfr to be considered neutral, and the two makers on chromosome 4, which were also included in the other data sets. All 97 blood samples were used.The third data set is from Bolivar State, Venezuela, a region of low transmission. It was described in The results are summarized in From The 95% profile-likelihood CIs for uela cf. . HoweverThe asymptotic confidence intervals agree well with the profile-likelihood CIs cf. –3. This Furthermore, we tested for pairwise differences between the estimates based on different marker loci. Overall, the methods perform well for all data sets and provide meaningful results. However, the statistical tests also yielded significant differences in some of the pairwise comparisons of the various h region –12. The Second, the tests are designed to compare the ML estimate based on the data with a value Third, the model assumptions might be violated, i.e., the underlying Poisson distribution might not be correct. This can again be quantified in the coarse of a robustness study.Fourth, the allele-frequency spectra of two different marker loci is very different, and the method might be sensitive to this. For instance strong skewness in the data distributions might bias the estimates. This is obviously the case if one marker shows no variation at all. Moreover, the number of different allele at different markers is very different, which results in very different probabilities of the ML estimates. These issues again need to be investigated in a numerical study.Fifth, some STR markers tend to be hyper-mutable. As a result, not just the frequency distribution might be more problematic, but it is also more challenging to correctly identify the tandem repeat numbers. Hence, for hyper-mutable markers the data might have very bad quality. In our examples the marker labelled L1 appears to be hyper-mutable.Because of all these possible reasons, it would be pre-mature to suggest a heuristic on how to decide, which estimates can be trusted the most. A systematic numerical follow-up study is planned to investigate all these possibilities in detail to provide suggestions on the criteria upon which the data is chosen.The number of genetically distinct lineages co-infecting a host - commonly referred to as “multiplicity of infection” (MOI) - is a key quantity in epidemiology. First, it relates with transmission intensity since it provides a metric for the number of secondary infections after a primary infection; assuming that the lineages circulating are identifiable . Second, it measures the possibility of genetic exchange among those lineages as determined by the genetic system of the pathogen in question. Finally, if phenotypic differences are associated with those lineages, MOI could lead to very complex dynamics driven by natural selection.Measuring MOI is desirable in a variety of infectious diseases, but - in many instances - only feasible if it can be measured at low cost and with a reasonable effort. Optimally it should fit into standard study designs and should be easily computable with whatever genotyping data can be collected from clinical specimens. In order to meet these goals, we further developed the maximum-likelihood (ML) method originally proposed by From a total of Although, our results justify the ML approach, it is nevertheless of fundamental importance to provide confidence intervals (CIs). We reported here on asymptotic and profile-likelihood-based CIs for all parameters. Asymptotic CIs are either based on the observed or the expected Fisher information, which under the CPD coincide. Explicit formulas for the CIs for all involved parameters were derived. Profile-likelihood based CIs were already emphasized by To test the ML estimate against other parameter choices typically three statistical tests are used, the likelihood-ratio, the Score, and the Wald test. The latter two are based on the asymptotic CIs, while the likelihood-ratio test builds upon the profile-likelihood-based CIs. Motivated by our intention to apply the methods to malaria we focused on using these tests to compare estimates for the parameter Not surprisingly all tests described perform equally well as they are asymptotically equivalent. However, as in the case of CIs we suggest to use the likelihood-ratio test if sample size is small or the parameters compared are small. If interested in p-values additional effort is required for the likelihood-ratio test, because a two-dimensional iteration needs to be performed. However, numerically this is only as demanding as obtaining the CIs. Because the test statistics for the Score and Wald tests can be derived, it is easy to derive p-values in these cases. For each of these two tests we provided three alternative variants, which all worked almost equally well in the provided examples. We should point out that it was our intention to indicate only how tests for the parameters can be constructed. With the usual approaches one could compare multiple parameters at the same time, including the information of all these markers. This however, exceeds both our intention and the scope of this article. Finally, as a justification for using the CPD, which simplifies the method to a great extent, we summarized the test suggested by Here, Assuming that the number of lineages infecting a host follows the CPD (2), the log-likelihood (3) simplifies to The proof of Remark 1 is as follows.Proof of Remark 1. First, note that Moreover, using de l'Hospitals rule we see that Assuming the CPD (2) the log-likelihood function is given by (4) and the derivatives of (5) are hence straightforwardly calculated to be The entries of the Hessian matrix (7), i.e., the second derivatives of First, the result showing existence and uniqueness of the ML estimate in the generic case is proven.Proof of Result 1. Assume By using (41) and equating (39c) to zero, we obtain Note that, The solution can be found by a Newton method. Because The special case, in which only single infections occur, is summarized by Remark 2. It can be proven as follows.Proof of Remark 2. Examining the proof of Result 1 yields that that the ML estimate is any positive root of Moreover, since In the limit In the other non-generic situation, every lineage is found in all samples, which is described in Remark 3 and can be proven as follows.Proof of Remark 3. The proof of Result 1 yields In this case the log-likelihood function simplifies to Taking the limit Since Remark 4 states that the expected number of samples containing a given lineage equals the observed number of samples containing this allele if the ML estimate is the true parameter. The proof is as follows.Proof of Remark 4. The maximum likelihood estimate satisfies In the following we will use that Successively repeating the last step gives Since the alleles can be arbitrarily labeled, we obtain The proof is completed by noting that The existence sand uniqueness of the profile-likelihood-based confidence intervals are proven as follows.Proof of Result 2. The proof consists of several parts.Existence in the generic case. We first assume Part A: The CI's bounds satisfy (12). The equations Hence, The likelihood function (4) can be rewritten as Note that Since Next, we show indirectly that First, assume Analogously it is shown that Since Uniqueness in the generic case. Next, the uniqueness of the confidence intervals is proven. Assume two values Part B: Existence and uniqueness in the non-generic cases. In the case Part C: Similarly, for Derivation of the CIs in the generic case. Parts A and B reveal that the bounds of the CI's bounds are the two solutions  and (16b).The Newton methods converges locally quadratically if the above matrix is nonsingular in the solutions. Part A of the proof reveals that these solutions satisfy Therefore, Clearly, since This proof is slightly more general than necessary as we will re-use part of it later.First, consider a matrix The formulae applies whenever, Hence, the desired quantity We are now ready to derive the confidence interval given by (18). To derive Moreover, Hence, Deriving By inspecting (40), it becomes clear that all derivations remain unchanged with  by  cf. . This giThe identities Substitution of the above evaluated at Proof of Result 4. To simplify the notation, we first derive the formulas for the confidence interval of We can apply a blockwise inversion formula to Clearly, Hence, the desired quantity To derive the desired quantity Therefore, Hence, Moreover, Combining the above yields, Hence, the bounds of the confidence intervals are given by By replacing Proof of Result 5. The result is proven by showing that the iteration (29) leads to the profile-likelihood with Remark 6.If and (and ) are the true (unknown) parameters, the asymptotic  holds.We aim to test only for , so any choice can be made for the true parameter. However, the parameters  occur in the asymptotic variance . Hence, we need a plug-in estimate for the asymptotic variance. There are two possibilities. First, the true parameter  is replaced by the profile-likelihood estimates  based on  and the asymptotic variance by . Here, either the expected or the observed Fisher information can be used.Second, both  and  can be replaced by the ML estimate . In this case the expected and observed Fisher information coincide.Proof of Result 6. The remark is proven by explicitly deriving the test statistic. To simplify the notation we write From the blockwise inversion formula (51) the relation Noting that Of course, (56) also holds if In the limiting case that the true parameter is Following the derivations in subsection 4.1, Remark 7.If the true parameter is , according to (57), an observation  with  is impossible in a sample of size . Hence,  with probability one.Assume As mentioned above, the case Hence, we can define The proof is found in the next subsection 6.2.From (59a) we immediately see that Applying de l'Hospitals rule gives Note, that the last steps also proves Similarly, from (39d) Since from (58)


Canine hip dysplasia (CHD) is characterised by a malformation of the hip joint, leading to osteoarthritis and lameness. Current breeding schemes against CHD have resulted in measurable but moderate responses. The application of marker-assisted selection, incorporating specific markers associated with the disease, or genomic selection, incorporating genome-wide markers, has the potential to dramatically improve results of breeding schemes. Our aims were to identify regions associated with hip dysplasia or its related traits using genome and chromosome-wide analysis, study the linkage disequilibrium (LD) in these regions and provide plausible gene candidates. This study is focused on the UK Labrador Retriever population, which has a high prevalence of the disease and participates in a recording program led by the British Veterinary Association (BVA) and The Kennel Club (KC).Two genome-wide and several chromosome-wide QTLs affecting CHD and its related traits were identified, indicating regions related to hip dysplasia.Consistent with previous studies, the genetic architecture of CHD appears to be based on many genes with small or moderate effect, suggesting that genomic selection rather than marker-assisted selection may be an appropriate strategy for reducing this disease.The online version of this article (doi:10.1186/1471-2164-15-833) contains supplementary material, which is available to authorized users. Canine hip dysplasia (CHD) is a pathology characterised by the malformation of the coxo-femoral joint, leading to degeneration of the hip joint, lameness and painful osteoarthritis. It mainly affects large-sized breeds like the German Shepherd and Labrador Retriever and has an important impact due to its high prevalence in these breeds  and The Kennel Club (KC) . In thiset al.  also obsCurrent breeding programmes against the disease are voluntary and recommend breeding from animals with HS  below a given threshold. They have resulted in measurable but moderate success –10. ApprThe purpose of this study was to perform genome- and chromosome-wide scans to identify QTLs for hip score and the three components associated with laxity in Labrador Retrievers, with the aims of improving selection schemes against the disease and furthering understanding of the biological basis of CHD.The genotyped sample comprised 1500 hip-scored Labrador Retrievers born between 2002 and 2008, which provides a representative sample of animals used for breeding. Dogs were evaluated for hip dysplasia based on radiographs according to the UK scoring method, recording the nine hip score components for each separate hip [https://www.thekennelclub.org.uk/services/public/mateselect/test/Default.aspx.).Radiographs were taken by veterinarians for submission to the British Veterinary Association/Kennel Club hip and elbow scoring schemes, a health screening protocol required before breeding from Kennel Club registered Labrador Retrievers. Hip and elbow score results are available to the public from the Kennel Club  was that no ethical approval was needed under the Animal Scientific Procedures Act (1986) because the technique was quick, non-invasive and painless and therefore was not a regulated procedure. The internal review process at the Institute also approved the research plan.Extraction of DNA from buccal swabs was performed according to a standard protocol . DNA wasQuality control was performed to assure both sample quality and marker quality . A samplFiltering criteria were applied to markers using Genome Studio software. A total of 59,260 markers were discarded due to low call rate (<98%), low reproducibility (GTS < 0.6), low or confounded signal (ABR mean < 0.3) and low minor allele frequency (MAF < 0.01). Further quality control on the markers was applied using PLINK , removinPreliminary analyses of population attributes  did not show associations with hip score, therefore, the genomic relationship matrix was fitted (as explained below) and was assumed to account for any remaining population structure.et al. [Association analyses were performed for total hip score (HS), transformed total hip score , following Lewis et al. ) and fory is the vector of phenotypes, W is a matrix of covariates with the α vector of associated fixed effects (including the intercept), x is a vector of marker genotypes (coded as 0/1/2) with β representing the regression of the phenotype on the marker genotypes, u is a vector of random polygenic effects and ϵ is a vector of residual errors. The fixed factors considered were the following: gender (1 degree of freedom), age at scoring in years and time spent exercising per day, which was scored from 1 (up to 1 hour) to 4 (more than 4 hours). The latter two factors were fitted as linear and quadratic regressions on deviations from their means. Analyses were conducted using GEMMA [G) [u, and errors, ϵ, follow multivariate normal (MVN) distributions given by u ~ MVN and ϵ ~ MVN, where VG and VE are the genetic variance associated with G and environmental variance, respectively.where ng GEMMA , which aEMMA [G) , and assGEMMA provides regression coefficient for each marker and their statistical significance was assessed using a Wald test. In determining a genome-wide significance threshold of P < 0.05, a conservative Bonferroni correction was made for multiple testing resulting from the large number of markers but not for multiple traits, resulting in a final threshold of P < 4.705E-7. The group of traits considered all share high pairwise genetic correlations, in excess of 0.85 , and the2 values) above their expectations under the null hypothesis. This inflation was detected and corrected by the use of the inflation factor λ, defined as the ratio of the median of the empirically observed distribution of the test statistic to the expected median , following the method suggested by Amin et al. [Differences in allele frequencies that are artefacts of cryptic population stratification or genotyping errors may inflate test statistics , where A is the numerator relationship matrix derived from the 5- generation pedigree of the phenotyped dogs, and VA is the associated additive genetic variance.where the meaning and distributional assumptions for each of these terms were as described above for the GWAS model. To compare the estimates of genetic variance obtained from genomic and pedigree analyses, the model was re-fitted with Using similar models the genetic variance was also partitioned among chromosomes using two different approaches: (1) a joint decomposition of the 38 canine chromosomes involving the simultaneous fitting of 38 chromosome-specific GRMs, or (2) chromosome-specific analyses in which the GRM for each chromosome was fitted together with a complementary GRM constructed from the all other SNPs to account for the remaining polygenic effect. In the first approach, the model fitted was of the form:y is the phenotype of each individual and, as in previous models, W and α refer to the fixed effects described above. In this approach , each u(i)  represents a vector of genetic values for chromosome i, such that u(i) ~ MVNG(i)) where G(i) is the GRM calculated from only those SNPs on chromosome i, and VC(i) is the associated variance. The estimate of total genetic variance is where u(i) is the effect of chromosome i, as described above, and distributed as MVNG(i)) and u(‒i) is a polygenic effect for the remaining genome and distributed as MVNG(-i)) where G(-i) is a GRM constructed from all SNPs other than those on chromosome i. Therefore in this second model, the model was fitted 38 times, once for each of the 38 chromosomes, with analysis of chromosome i providing an estimate for VC(i).where VG[VG across the entire genome was also evaluated. In the latter analysis, the genome was divided into distinct regions, each containing 20 adjacent SNPs. The analysis then proceeded similarly to the approach described above for chromosomal variances with two components. Under the first approach, the SNP windows were always centred at the GWAS-significant SNPs, whereas under the second one, the significant SNP was not necessarily at the centre of a window.Regional VG for windThe regional variances for significant SNPs and genomic and chromosomal variances were estimated for each trait using ACTA , but theAn additional analysis of the haplotype structure and linkage disequilibrium (LD) in the genome-wide significant regions was performed using Haploview 4.2 . Genome-VE) and genetic (VG) variances for each trait are shown in Table VC) at the chromosome level calculated using the two different approaches described above: joint chromosomal decomposition of the 38 chromosomes . In contrast, for THS, VC was more evenly distributed across the genome, with Chr 1, 9, 21 and 24 explaining the highest proportions of VC, 6.6%, 15.3%, 11% and 8.3%, respectively  from genomic analyses were greater (and heritability lower) than their pedigree estimates  whereas for the left-hip NA, the total VG was more evenly distributed across the genome, with several chromosomes explaining similar percentages. Results for total NA were a mixture of those for left and right hips, with Chr 3, 4, 9, 11 and 21 explaining the highest proportions of VC, 6.7%, 6.9%, 6.8%, 7.5% and 6.7%, respectively  from genomic analyses for all three traits associated with NA were greater (with lower heritability) than the pedigree estimates (Table VG) but the standard error was very high . In the case of SUB total, Chr 1, 9 and 24 explained the highest proportions, 7.2%, 7.2% and 7.3% of the total VG, respectively  from genomic analyses for all three traits were again slightly higher (lower heritability) than the pedigree estimates  for the three traits were very close to (and slightly lower than) pedigree estimates (Table VG) but again the standard errors were very high , support this prediction and therefore, we subsequently focus on estimates of individual chromosomal variance from the joint chromosomal decomposition. These estimates revealed a polygenic architecture for all traits, with several chromosomes contributing less than 6% of the total genetic variance and a few chromosomes contributing 15-23%.Due to the Beavis effect, the estimates of effect sizes (Beta coefficients) and the proportion of genetic variance explained by each chromosome or region, considered separately, are expected to be overestimates . Therefoet al. [All animals, including those with perfect hip scores (HS = 0), were included in the analyses. However, Lewis et al.  detectedThe present study used a high density canine chip to identify two genome-wide QTLs affecting CHD-related traits. Chr 1 and 21 revealed consistent GWAS peaks and significant SNPs across the analyses in relation to several traits. Other suggestive QTLs were detected as significant at the chromosome-wide level, and may also indicate genomic regions related to hip dysplasia. In order to strengthen the evidence for QTLs identified in our study, we have applied two additional analyses: chromosomal and regional analyses of the explained genetic variance. These methods are designed to estimate variances associated with multiple loci contributing localised variance that may be too small individually to detect by single association tests , 33 and VG observed for CrAE left and CrAE total, respectively, and revealed six genome-wide significant SNPs for these traits, as well as chromosome-wide significant SNPs related to HS and NA right. These SNPs were located between two QTLs for CHD status in German Shepherds [SHC3 gene, which is not known to be associated with CHD-related traits, however, within 1 Mb of this gene, there were several functional candidate genes, associated with bone formation or mineralization , cartilage formation (PHF2), and differentiation into joint versus cartilage cells (BARX1). However, none of the SNPs within or closest to these genes showed high LD (assessed by r2) with the SNPs identified by GWAS. The third genome-wide significant SNP was found in an inter-genic region, close to a zinc-finger gene (ZNF677), which again is not known to have any association with CHD-related traits, and several uncharacterised protein-coding genes. The large size of the Chr 1 region, and hence the large number of positional candidate genes, highlights a problem in the identification of causative variants from GWAS studies in species with high levels of LD such as dogs.Chr 1 explained ~23% and ~13% of the total hepherds  and withhepherds . Genes ir2 (and power of detection) between a marker and a causal variant is sensitive to differences in allele frequency, particularly at low frequencies, suggesting that the allele frequencies of the marker and causative variant cannot be radically different.While this 11-Mb region on Chr 1 included the greatest number of genome-wide significant SNPs in our study, the frequency of the favourable SNP alleles (associated with low hip score) is already very high, suggesting that the average effect of the favourable allele is small  and consVG observed for NA right, revealing one genome-wide significant SNP for this trait, several chromosome-wide significant SNPs related to NA, CrAE and HS and a significant peak in the regional analysis for THS. These SNPs correspond to two main regions: a genome-wide significant region situated at ~43 Mb and a chromosome-wide significant region situated at ~27 Mb. For the first case, previous studies have shown the existence of a QTL related to NA in this region for a Labrador-Greyhound cross-breed pedigree [OTOG gene, which encodes an ear-specific glycoprotein that has no known association with CHD-related traits. However, as with the Chr 1 region, there are several functional candidate genes between 41.5 and 44.5 MB, the region that includes the genome- and chromosome-wide significant SNPs. These include genes associated with cartilage formation (SOX6), inflammation and osteoarthritis (SAA) and muscle cell differentiation and muscle regeneration . Again, most of the SNPs within or closest to these genes showed low LD with the SNPs identified by GWAS, however, one SNP within SERGEF was strongly correlated (r2 = 0.8) with the genome-significant SNP and two SNPs within SOX6 were moderately correlated with it (r2 = 0.4). In comparison to the region on Chr 1, there is greater potential for genetic progress based on this region as the frequency of the favourable allele is much lower for this QTL.Chr 21 explained ~17% of the total pedigree . The genChromosome-wide QTLs were detected on several chromosomes and a polygenic pattern was also observed for the distribution of the genetic variance in both the regional and chromosomal studies, with most of the chromosomes explaining less than 6% of the genetic variance but some chromosomes explaining between 10% and 23%. These results confirm a polygenic nature of the disease that could arise if the hip score and its components were the result of multiple interacting processes. It might be the case that genes affecting one or two processes have a small effect on the main phenotype , while only genes affecting several processes appear as QTLs with larger effects. The complex aetiology of the disease supports its polygenic nature. Both biological and environmental factors potentially contribute to an unstable hip joint  and manyAnother aspect of the trait complexity found in our study and several others , 17, 36 VE). Residual variance was used to compare the models because in both cases, it comprises the variance component not explained by the model for the phenotypes in the data, whereas, in contrast, the genetic variances obtained using genomic and numerator (pedigree-based) relationship matrices refer to constructed base populations, which may differ. SNP-based methods are not expected to capture all the genetic variance [VE greater than those provided by a pedigree analysis for most of the traits, but not in all cases. It is expected that an increase in the number of markers will increase the proportion of explained genetic variance [In order to further explore the QTLs identified in our study, the performance of the SNP-based genomic approach to characterise the genetic architecture of the trait was also compared with the pedigree approach in terms of residual variance  correspond with those found in previous QTL studies, others have not been seen before and equally, we did not detect effects in several previously identified regions. There was no overlap between the regions showing genome- or chromosome-wide significance and those associated with CHD severity in a recent case-control study of 78 Dutch Labrador Retrievers  , associated trait, minor allele and its frequency, Beta coefficient , P-value of the GWAS analysis and percentage of genetic variance explained by a region of 21 SNPs centred in the significant SNP. (XLSX 19 KB)Additional file 1:Quantile-Quantile plots for GWAS analyses. Figures show the Q-Q plots after correcting by the inflation factor. (PDF 301 KB)Additional file 2:Percentage of genetic variance explained per chromosome: The joint chromosomal decomposition model fitted the 38 chromosomes simultaneously. . (PDF 1 MB)Additional file 6:Percentage of genetic variance explained for all significant SNPs. Contains the percentage of genetic variance explained by all significant SNPs (% VG SNPs) or by the regions of 20 SNPs centered at the significant SNPs (% VG Regions). (XLSX 11 KB)Additional file 7:


The future of medicine is moving towards the phase of precision medicine, with the goal to prevent and treat diseases by taking inter-individual variability into account. A large part of the variability lies in our genetic makeup. With the fast paced improvement of high-throughput methods for genome sequencing, a tremendous amount of genetics data have already been generated. The next hurdle for precision medicine is to have sufficient computational tools for analyzing large sets of data. Genome-Wide Association Studies (GWAS) have been the primary method to assess the relationship between single nucleotide polymorphisms (SNPs) and disease traits. While GWAS is sufficient in finding individual SNPs with strong main effects, it does not capture potential interactions among multiple SNPs. In many traits, a large proportion of variation remain unexplained by using main effects alone, leaving the door open for exploring the role of genetic interactions. However, identifying genetic interactions in large-scale genomics data poses a challenge even for modern computing.INADL and LPP that are known to be associated with T2D.For this study, we present a new algorithm, Grammatical Evolution Bayesian Network (GEBN) that utilizes Bayesian Networks to identify interactions in the data, and at the same time, uses an evolutionary algorithm to reduce the computational cost associated with network optimization. GEBN excelled in simulation studies where the data contained main effects and interaction effects. We also applied GEBN to a Type 2 diabetes (T2D) dataset obtained from the Marshfield Personalized Medicine Research Project (PMRP). We were able to identify genetic interactions for T2D cases and controls and use information from those interactions to classify T2D samples. We obtained an average testing area under the curve (AUC) of 86.8 %. We also identified several interacting genes such as Developing the computational tools to explore genetic associations beyond main effects remains a critically important challenge in human genetics. Methods, such as GEBN, demonstrate the utility of considering genetic interactions, as they likely explain some of the missing heritability. Over the past decade, development in large-scale, high-throughput methods to characterize the human genome has dramatically improved our ability to assess the relationship between an individuals’ genome and diseases . With thMuch variability in human diseases and traits remain unexplained by using GWAS alone . It is hTo capture main effects of genetic variants as well as complex genetic interactions, we created the Grammatical Evolution Bayesian Network (GEBN) algorithm. The algorithm can simultaneously identify marginal effects as well as interaction effects without exponentially increasing the search time. GEBN can also identify interactions that occur between different sets of genetic variants in different groups (i.e. cases and controls). This flexibility allows discovery of non-overlapping genetic architectures in multiple groups. Previous Bayesian Networks methods to detect genetic interactions , 14 haveWe implemented GEBN algorithm in the software package ATHENA. We tested the algorithm on various simulation datasets. We also applied GEBN to a case–control dataset for type 2 diabetes obtained from the Marshfield Personalized Medicine Research Project Biobank (Marshfield PMRP) . The netDivide the data into five equal parts for cross-validationsFor each cross validation:Populations of binary string are randomly generated and translated into functional Bayesian Networks by the grammar. For each individual genome, the binary string is divided into consecutive codons. The codons are then translated according to the grammar Fig. .Fig. 1GeCalculate the fitness of the Bayesian Networks using the K2 scoring function .i is the number of different values of Xi’s parents, ri is the number of values of Xi. The score calculates the probability of observing the network given the data.Where D is the dataset, B is Bayesian Network, n is total number of variables, qSelect the Bayesian Networks that have the highest fitness, which will then undergo crossover and mutations. During crossover and mutation, parts of the different Bayesian Networks are exchanged or mutated to create new networks.Repeat 3–4 for a set number of generationsSave the best model in the final generation and evaluate it on testing dataBayesian Network is a multivariate modeling method that expresses the relationship of variables through a series of conditional distributions. The use of Bayesian Networks is becoming very important in biology because of their ability to infer biological networks , model sThe final Bayesian Network is composed of connected and unconnected variables. Variables that are connected in the network are directly dependent with each other, while unconnected variables are conditionally independent. The advantage of GEBN over the more traditional network construction is that it can explore a wider search space, thus more suitable for large-scale genomics data. In addition, using an evolutionary search strategy removes the dependency on human trial and error to create optimal network structures and instead relies on the data and computation along with evolutionary learning to find optimal structures.The above GEBN method is applied to the case group and the control group independently. To prevent over-fitting, we used Bayesian Information Criteria (BIC)  to contrWhere L is the maximum likelihood of data given a network, k is the number of free parameters, and n is the sample size. We iteratively removed each edge in the case or control network and calculated BIC for the reduced model. If the reduced model had higher BIC value, the edge was retained, and vice versa.Finally, we used the discriminant analysis to assign an individual into either the case group or the control group. Using Bayes theorem, the probability of the sample belonging to a case group is calculated by:Where p = total number of variables. To test our approach, we simulated data that contains functional SNP variables with main effects and interaction effects. For main effect simulation, we simulated data that consist of different numbers of functional SNPs with varying degrees of association to a binary outcome. For interaction effects, we separately simulated a number of interaction effects in case and control groups. We purposely made the interaction effects different in case and control groups to mimic different genetic architectures in two groups Fig. .Fig. 2Sccase) using a weighted average of allele frequencies in the control data (Fcontrol) and the extreme allele frequencies (Feffect) that were defined as . Thus, the allele frequencies of the functional SNP in the case data is obtained by Fcase = w*Feffect + (1-w)*Fcontrol, where w is the weight index – larger w indicating more discrepancy between the case frequencies and control frequencies.To simulate different degrees of main effect, for a functional SNP, we altered the allele frequencies in the case data * Find.The interaction effects were simulated as follows: Let FFor each dataset, we used the simulated frequency tables with sampling with replacement to determine the genotype of the functional SNPs. Then, we embedded the functional SNPs into a dataset with random SNPs to make it comparable to real biological datasets. Details of simulation parameters are shown in Table The Marshfield PMRP is a biobank that has collected ~20,000 adult subjects’ biological samples and electronic health records . We obtaIn the simulation study, we compared the performance of GEBN to that of the traditional GWAS approach based on logistic regression and another widely used method for detecting interactions, grammatical evolution neural network (GENN) , 28. TheFor main effect models, GEBN achieved close to maximum prediction performance in datasets with 100 SNPs. Logistic regression showed similar power, while GENN showed lower power. With 500 SNPs, the performance advantage of GEBN is even more visible Fig. a-d. The When case data and control data only differ by SNP interactions, logistic regression failed to separate two types with ROC curve fluctuating along the 45° line which corresponds to random guesses. GENN showed some power in detect interactions. However, GEBN showed improved ROC especially when the effect size is large Fig. e-h. The p < 0.001 as threshold, we identified 259 SNPs associated with type 2 diabetes. The top associated SNP was rs7903146 (p = 2.997e–06), which maps to TCF7L2 gene. To remove SNPs that are correlated, we used PLINK software [We first performed association analysis using logistic regression for 267,209 SNPs associations with type 2 diabetes, using software  to pruneP3H2 gene), rs2666504 and rs1475563 (INADL gene). There was also one unique interaction for cases, which is rs10065876 and rs11741322 and two for controls, which are rs4477348 and rs6480213 (both in CTNNA3 gene), and rs11707430 and rs6444295 (both in LPP gene).Figure In this study, we presented a novel algorithm that can efficiently capture marginal and interaction effects present in the genetic data. We demonstrated in simulation data that GEBN performed equal or better than the standard GWAS analysis method using logistic regression as well as GENN on data with only main effect functional SNPs. In data with interacting SNPs, logistic regression failed to capture the true model which is shown by the ~50 % AUC Fig. . GENN waTCF7L2 gene. We also identified rs12255372, which is in LD with rs7903146, as a significant association. TCF7L2 gene has been implicated for type 2 diabetes in many studies [P3H2 gene and one SNP in INADL gene and one SNP in the non-coding region of chromosome 1. The INADL gene is part of the hippo signaling pathway [INADL was associated with children’s weight [CTNNA3 were found to be associated with Alzheimer [LPP gene has shown a robust association with type 2 diabetes in multiple ethnicities as well as combined meta-analysis [Using main effect filtering followed by GEBN analysis, we replicated canonical associations and also identified novel genetic interactions for type 2 diabetes. The most significant association was rs7903146, which is located in the  studies , 31. We  pathway . The pat pathway  and adip pathway . Interess weight . It is dlzheimer  and hearanalysis . Taken tThis paper presents the first step of the algorithm development that aims to address the pressing need for tools to identify complex relationships within the genetics data. Due to the flexibility of the Bayesian networks, the algorithm could be applied to datasets with more than two outcomes. For example, drug response phenotypes might be categorized as high responder, low responder, and non-responder. This would be possible to analyze with GEBN.The utility of GEBN will be even greater in those settings because traditional statistical approaches are generally limited to binary outcomes. We also plan to integrate other –omics data such as transcriptomic and methylomic data into the network. The potential interactions between factors from different data types could reveal novel biological insights not seen at any individual data alone. The ultimate goal of individually identifying networks for different groups or subtypes of disease is to more precisely understand the disease so that we can improve detection and treatment of the disease. The method presented in this paper will help further elucidate the complex biological relationship present in the genetics data.