Motivation: In order to discover quantitative trait loci, multi-dimensional genomic datasets combining DNA-seq and ChiP-/RNA-seq require methods that rapidly correlate tens of thousands of molecular phenotypes with millions of genetic variants while appropriately controlling for multiple testing.Results: We have developed FastQTL, a method that implements a popular cis-QTL mapping strategy in a user- and cluster-friendly tool. FastQTL also proposes an efficient permutation procedure to control for multiple testing. The outcome of permutations is modeled using beta distributions trained from a few permutations and from which adjusted P-values can be estimated at any level of significance with little computational cost. The Geuvadis & GTEx pilot datasets can be now easily analyzed an order of magnitude faster than previous approaches.Availability and implementation: Source code, binaries and comprehensive documentation of FastQTL are freely available to download at http://fastqtl.sourceforge.net/Contact:emmanouil.dermitzakis@unige.ch or olivier.delaneau@unige.chSupplementary information: Supplementary data are available at Bioinformatics online. Matrix eQTL , the Bonferroni method usually proves to be overly stringent and results in many false negatives. To overcome this issue, a commonly adopted approach . Then, we can easily assess how likely an observed association obtained in the nominal pass originates from the null, resulting in an adjusted P-value. In practice, performing permutations in this context requires fast methods able to absorb such substantial computational loads in reasonable running times. Even though Matrix eQTL has been used so far in multiple large-scale studies , we model this null distribution of most significant P-values for a phenotype with a beta distribution, learning the parameters from a few permutations  by maximum likelihood estimation. As a result, we obtain a reasonably good approximation of the tail of null distribution to estimate small adjusted P-values at any significance level (i.e. without lower bound). In a final stage, a false discovery rate (FDR) procedure as implemented in the R/qvalue , (2) or (3) to extract all significant phenotype-variant pairs at a given FDR, usually chosen to be 5% or 10% (see Section 2.6). All this, plus other optional functionalities, have been implemented in the FastQTL software package (see Section 2.7).FastQTL performs linear regressions between genotypes and molecular phenotypes with or without covariates in order to find the best nominal association for each phenotype (see Section 2.2). Then, it can correct for the multiple correlated variants tested via three different permutation schemes: (1) a direct permutation scheme that relies on a fixed number of permutations (see Section 2.3), (2) an adaptive permutation scheme which maintains a reasonable computational load by tailoring the number of permutations to the significance of the association (see Section 2.4) and (3) a beta approximation which models the permutation outcome via a beta distribution (see Section 2.5). For (1) and (2), an adjusted R/qvalue  package P quantified in a set of N samples. Let G be the set of genotype dosages at L variant sites located within a cis-window of ± W Mb of the genomic location of P. To discover the best candidate QTL for P, FastQTL measures Pearson product-moment correlation coefficients between P and all L variants in G, stores the most strongly correlated variant q ∈ G as candidate QTL, and assesses its significance by calculating a nominal P-value np with standard significance tests for Pearson correlation. Note that this is equivalent to testing for β≠0 in a linear model P = βg + ϵ with β estimated by least squares fitting. Of note, this method is also used by Matrix eQTL to speed up linear regression  and FDR estimation to control for the latter (see Section 2.6).For simplicity, we will focus on a single molecular phenotype P-values is very difficult due to the fact that we test multiple variants per phenotype, correlated because of LD, across a wide allele frequency spectrum, while all this varies from one phenotype to another. To account for this, significance of a candidate QTL is assessed via permutations. Specifically, we repeat the cis-window scan procedure for R random permutations of P, leaving the genotype data G unchanged to preserve the correlation between variants. Each time, we store the strongest correlation; the goal is to produce a sample from the distribution of the strongest correlation under the complete null hypothesis of no genetic associations. Then, the observed correlation is compared with this empirical null distribution to obtain an adjusted P-value characterizing the significance of the candidate QTL. When very few null correlations are found to be stronger than the observed one, it means that reaching this correlation level by chance is very unlikely and therefore that the QTL candidate is likely to be true. More formally, if r correlations in the null distribution are found to be stronger than the observed, significance of the QTL candidate is assessed by calculating the following empirical adjusted P-value dp of association ; meaning that a large number of permutations are needed to get precise estimates of small adjusted P-values, thereby increasing the computational burden. For instance, reaching P-values of ∼10−3 requires thousands of permutations while billions are needed to get P-values of ∼10−9. In practice, it is very difficult to go beyond a few thousand permutations genome-wide with this approach, which forces us to work with adjusted P-values in the range of 10−3 to 1.0. To alleviate this limitation, we improved the direct permutation scheme with two complementary methods (see Sections 2.4 and 2.5).The choice of a proper global significance threshold for nominal ociation :(1)pd=rP-values can be achieved with few permutations while many more are needed to estimate highly significant ones. Therefore, we implemented an alternative permutation scheme that adapts the number of permutations to the significance level of the variant–phenotype pairs  of null correlations stronger than the observed one are found. To prevent this strategy running too long for the most significant variant–phenotype pairs, the algorithm cannot perform more than M  permutations in total. Then, an adjusted P-value of association for a candidate QTL is derived using:From P-values below 1/(M+1), though M can scale up to 100 000 in practice and thus provide good estimations for adjusted P-values down to ∼10−5.This strategy still remains unable to provide adjusted P-values at any significance level without actually performing all required permutations, we developed an approximation method based on the beta distribution. It is well established that order statistics of independent uniformly distributed random variables are beta-distributed , the L tests performed are not independent, implying that the effective number of tests n is lower than the actual number L of variants in cis. Instead of fixing the k and n parameters a priori, we use a more flexible approach in which the parameters are estimated by maximum likelihood  fitted beta distribution with:Note that this maximization is done using standard numerical methods implemented in GNU Scientific Library. The underlying idea of this approach is to characterize the extreme tail of the null distribution without directly sampling from it, something that would entail a huge computational burden. Finally, we can approximate an adjusted π0, defined as the proportion of hypotheses for which the null is true. Implicitly, the BH procedure assumes π0 is 1, whereas the ST procedure learns it from the data, resulting in more statistically significant hits. Of note, the adjusted P-values provided by FastQTL allow the users to easily apply whichever multiple testing correction they favor, from FDR to Bonferroni, since it provides adjusted P-values well calibrated on the full P-value range.Since thousands of molecular phenotypes are tested genome-wide, an FDR correction is commonly applied. This estimates the proportion of false positive findings, known as the FDR, by comparing the number of hits declared to the number that would be expected by chance. The Benjamini–Hochberg (BH) procedure  is one wP-value per phenotype. An FDR procedure (see Section 2.6) is then straightforward to apply on the FastQTL output. In addition to the functionalities described above, FastQTL also implements some additional useful features worth mentioning here:FastQTL implements in C ++ (see Sections 2.1–2.5) to provide an adjusted R/rntransform function of the GenABEL package ((1) To make the method more robust to outliers in the phenotype data, FastQTL has an option that allows to quantile normalize the phenotype quantifications prior to any analysis. This ensures that phenotype quantifications are normally distributed with mean 0 and standard deviation 1. Quantile normalization is implemented as in the (2) Confounding factors such as population stratification and experimental batch effects have to be considered to prevent spurious associations. To do so, FastQTL can residualize both the genotypes and the phenotypes for quantitative and/or qualitative covariates prior to any association testing.(3) FastQTL uses standard file formats: genotype dosages and phenotype quantifications are specified in Variant Calling Format and UCSC BED format, respectively. All files are required to be indexed with Tabix  to enablcommand 1.(4) To split a genome-wide analysis into non-overlapping chunks and to run each on a distinct CPU core, FastQTL includes a set of user-friendly options. It can either split the data into a given number of chunk (--chunk 12 200 to run chunk 12 out of 200) or focus on a particular user defined genomic region (--region 20:1-1000000). The phenotype and genotype data included in the genomic region is then automatically extracted from the cis-window size and analyzed. A simple loop going through all possible chunks allows the user to submit the full analysis on a compute cluster or server. A FastQTL example command line to perform a genome-wide analysis is shown in (1)   for c in $(seq 1 256); do(2)             fastQTL  --vcf genotypes.vcf.gz(3)                        --bed phenotypes.bed.gz(4)                        --chunk $c 256(5)                        --permute 1000(6)                        --output results.$c\.txt.gz(7)    done(8)    zcat results.*.txt.gz | gzip -c > results.txt.gzCommand1. This shows the BASH script needed to run a genome-wide analysis. The genotypes and phenotypes are specified with --vcf (line 2) and --bed (line 3), respectively. The analysis is split into 256 non-overlapping chunks (lines 1 and 4) and is based on 1000 permutations (line 5). The full outcome is constructed by concatenating the per-chunk outcomes (line 8).Supplementary material 1) and the GTEx consortia , two of the largest eQTL studies performed to date. This comprises a total of 10 distinct datasets with between 14 and 35K quantified genes and 6.8–10.8M variant sites for 83–373 samples (Supplementary Table 1).To perform a comprehensive evaluation of FastQTL, we used RNA-seq and genotype data produced by both the Geuvadis  stored for each phenotype the best P-values obtained from 1000 permutations as observations, (2) estimated k and n by ML from the 1000 resulting P-values, (3) simulated 1000 P-values from the newly parameterized beta distribution as expectations and (4) compared both observations and expectations to assess their goodness-of-fit visually (QQ-plots) and statistically (one sample Kolmogorov–Smirnov test). Overall, we find very high degrees of concordance between results; both when pooling all genes together . Of note, the beta approximation provides small adjusted P-values that are better calibrated than those provided by the direct method  and sometimes not even accessible (i.e. below the lower bound implied by the number of permutations); the smallest adjusted P-value estimated using the GEUV_EUR dataset is in the order of ∼10−128 . Therefore, we subsequently estimated the number of permutations that the direct method needs to reach the same level of calibration as the beta approximation at various significance levels. To do so, we binned the adjusted P-values obtained from beta approximations and estimated for each bin, by exhaustive search, the number of permutations required by the direct method to match the same sampling variation . We find that this number drastically increases as small-adjusted P-values are targeted . As a consequence, using 500 permutations with the beta approximation, for example, has the same accuracy to recover the optimal set as the direct method using 1000 permutations. Interestingly, the beta approximation using only 50/100 permutations already does very well at recovering the optimal eQTL set.Supplementary material 4). Overall, we find that both FastQTL and Matrix eQTL recapitulate the official eQTL set well, especially as the number of permutations increases  and FastQTL (beta approximation) using 100, 500 and 1000 permutations could reproduce the outcome of the pilot phase of GTEx; a large-scale eQTL mapping study. Of note, we run Matrix eQTL in the most highly effective setting we could achieve in order to fully utilize its matrix based design  is of crucial importance to (1) estimate the number of tests made under the null (quantity underlying efficient FDR correction methods) and to (2) meta-analyze multiple QTL studies together. Second, the beta approximation behaves well enough with only 100 permutations to rapidly assess the impact on the analysis of important parameters such as cis-window size and covariates like the number of PEER factors (P-values than the best method so far, Matrix eQTL with a direct permutation scheme, while being significantly faster. In addition, FastQTL also provides a modular base onto which new functionalities are being implemented, such as fine mapping of causal variants, conditional analysis to discover multiple independent QTLs per phenotype and interaction analysis to discover sex or disease specific QTLs.We present FastQTL, a QTL mapper in cis for molecular phenotypes that implements a new permutation scheme to accurately and rapidly correct for multiple-testing at both the genotype and phenotype levels. FastQTL has several advantages compared with existing methods, making it the ideal candidate to map QTLs for the coming wave of large-scale datasets regrouping many different layers of molecular phenotypes and near complete collection of variant sites. First, permutations are modeled with a beta distribution, parameterized from a relatively small number of permutations. This results in accurate adjusted


Current sequencing technology enables generation of whole genome sequencing data sets that contain a high density of rare variants, each of which is carried by, at most, 5% of the sampled subjects. Such variants are involved in the etiology of most common diseases in humans. These diseases can be studied by relevant longitudinal phenotype traits. Tests for association between such genotype information and longitudinal traits allow the study of the function of rare variants in complex human disorders. In this paper, we propose an association-screening framework that highlights the genotypic differences observed on rare variants and the longitudinal nature of phenotypes. In particular, both variants within a gene and longitudinal phenotypes are used to create partitions of subjects. Association between the 2 sets of constructed partitions is then evaluated. We apply the proposed strategy to the simulated data from the Genetic Analysis Workshop 18 and compare the obtained results with those from sequence kernel association test using the receiver operating characteristic curves. Rare variants have been speculated to be involved in the etiology of complex human diseases . Such diMany genetic variants are rare variants . Because of their low MAFs, traditional association methods may suffer from low power. A natural idea for improving power is grouping or collapsing together certain variants. Such collapsing methods are based on the assumption that rare variants in a group  may function in combination . For exaLongitudinal traits  provide valuable information regarding the progression of diseases. Traditionally, such longitudinal data can be analyzed using the so-called cross-sectional strategies. In particular, such methods involve repeating the same analysis at various, specific points in time. Since at each time point the trait under consideration reduces to a scalar, methods such as inverse-probability clustering can be conducted for association screening. Then, variants can be selected based on the results from each time point. The assumption underlying this type of strategy is that genetic variants maintain similar influences at different time points. However, it is more likely that those variants influence the pattern of the traits across time; for example, a group of variants may affect how blood pressure changes in a time-dependent manner. Cross-sectional analysis may fail under such circumstances. A method that takes full consideration of the longitudinal nature of traits is thus desired to capture such genetics-time interactions.In this paper, we propose a dual-clustering framework, which highlights both rare variants and the longitudinal structure of traits. By "dual" clustering, we mean individuals are clustered based on both genotypic information through inverse-probability weighting and longitudinal traits through ordinary hierarchical clustering. Association between the 2 sets of partition labels can then be readily evaluated using existing single-marker and scalar-trait association methods, such as one-way analysis of variance (ANOVA) or the partition retention (PR) method ,6. We apThe simulated data set of GAW18 is a combination of real WGS data and simulated longitudinal traits. The sequence data is drawn from T2D-GENES Project 2. In this paper, we use the dosage genotype data on chromosome 3, which include 773,088 single-nucleotide polymorphisms (SNPs) that can be mapped to the genome. Two hundred phenotype sets were simulated based on genotype data. For each simulated data set, we analyze systolic blood pressure (SBP) and diastolic blood pressure (DBP), each with measurements at 3 time points, for 849 related subjects. We map the SNPs to its host gene, resulting in 1426 genes.ik g= 0, 1, or 2 represent the number of minor alleles at SNP k for individual i, and let k p be the observed MAF of SNP k. We define the inverse-probability weighted similarity score between individuals i and j based on SNP k as:Let G, the similarity between individuals i and j is defined as the sum of the similarity scores on SNPs within that gene: i, j)'s, are evaluated first and then converted to a distance measure using the transformation The definition in equation (1) highlights the influence of rare variants, and the genotypic similarity on minor alleles, but not that on major alleles . For a gous work , and parResults).The main difficulty of dealing with longitudinal traits is that most existing association methods only consider scalar phenotypes. Thus it is natural to transform longitudinal traits into some 1-dimensional summary statistics. Here we adopt ordinary hierarchical clustering using phenotype vectors and treat the resulting class labels as a summary statistic. Because hierarchical clustering uses the whole longitudinal trait as features, we expect that it can capture the structure contained in the phenotypes. In this study, we cluster the 849 individuals into 2 groups. Results show that these 2 groups can be treated as with high and low blood pressures  are defined based on the corresponding MAFs, such that the influence of rare variants can be boosted, an idea morally similar to the similarity scores defined in equation (1). For a particular gene, similarity between 2 individuals can be defined by the same summation as in our method.We also analyze the data using the linear SKAT  for compSKAT uses the variance-component score statistic based on the above similarity scores to test for association between genotypic variants and a scalar trait. We treat the cluster indices from the longitudinal traits as the response variable in order to apply SKAT to GAW18 data. More details on SKAT can be found in Ref. .We first apply the proposed method to the WGS dosage data including all the 773,088 SNPs and the SBP trait. Three genes are discovered after Bonferroni correction, of which 1 gene, Y_RNA, is significant in 15 of the 200 replicates. It turns out that this gene resides within MAP4, which has the strongest signal in the simulated model, and produces a noncoding RNA.I has a bigger area under curve (AUC). Results are similar for other partition sizes resulted from inverse-probability clustering.One reason for the relatively few significant genes obtained above may be that there is a very high density of variants within most genes. We then conduct similar analysis using only SNPs with MAFs between 0.01 and 0.05 to increase power. SBP and DBP are regressed on age, sex, age × sex, and medication, and the residuals are used in the clustering analysis. For method comparison, we treat genes containing at least 1 causal SNP in the simulated model as causal genes, resulting in 21 genes for SBP and 26 genes for DBP. We compare the receiver operating characteristic (ROC) curves by the proposed dual-clustering framework and SKAT  for both causal and noncausal variants. Therefore, the main conclusion based on ROC curves is still valid. In practice, we suggest evaluating p values using permutations and controlling the false discovery rate in order to have better sensitivity to real genetic signals. This may introduce more computational burden, but it is worth mentioning that the 2 clustering tasks can be done independently and simultaneously so that the computational time can be reduced. Multilevel models with Markov chain Monte Carlo (MCMC) techniques may also address the multiple comparisons problem encountered here by partial pooling [Although the simulation model did not take family structures into account, the ANOVA  pooling .The methods we experimented on have relatively low power on this particular data set. Our framework obtains slightly better results in terms of AUC. It is worth applying the proposed methods to other data sets for a comprehensive understanding of its performance.The authors declare that they have no competing interests.TZ and YL conceived of and designed the study, performed the statistical analysis, and drafted the manuscript. TZ participated in its coordination. TZ, YL, CHH, SHL and IH discussed the results. All authors read and approved the final manuscript.


Genome-wide association analysis in tropical and subtropical maize germplasm revealedthatMLND resistance is influenced by multiple genomic regions with small to medium effects.Maize chlorotic mottle virus and Sugarcane mosaic virus, and has emerged as a serious threat to maize production in eastern Africa since 2011. Our objective was to gain insights into the genetic architecture underlying the resistance to MLND by genome-wide association study (GWAS) and genomic selection. We used two association mapping (AM) panels comprising a total of 615 diverse tropical/subtropical maize inbred lines. All the lines were evaluated against MLND under artificial inoculation. Both the panels were genotyped using genotyping-by-sequencing. Phenotypic variation for MLND resistance was significant and heritability was moderately high in both the panels. Few promising lines with high resistance to MLND were identified to be used as potential donors. GWAS revealed 24 SNPs that were significantly associated (P < 3 × 10−5) with MLND resistance. These SNPs are located within or adjacent to 20 putative candidate genes that are associated with plant disease resistance. Ridge regression best linear unbiased prediction with five-fold cross-validation revealed higher prediction accuracy for IMAS-AM panel (0.56) over DTMA-AM (0.36) panel. The prediction accuracy for both within and across panels is promising; inclusion of MLND resistance associated SNPs into the prediction model further improved the accuracy. Overall, the study revealed that resistance to MLND is controlled by multiple loci with small to medium effects and the SNPs identified by GWAS can be used as potential candidates in MLND resistance breeding program.The maize lethal necrosis disease (MLND) caused by synergistic interaction of The online version of this article (doi:10.1007/s00122-015-2559-0) contains supplementary material, which is available to authorized users. Maize chlorotic mottle virus (MCMV) and Sugarcane mosaic virus (SCMV). Although each of these viruses individually can cause disease, the synergistic interactions are more pronounced. SCMV was reported in Kenya many years ago  has emerged as a devastating disease in eastern Africa since 2011  enables analysis of genetic architecture of complex traits  is another promising breeding tool to improve the efficiency and speed of the breeding process  panels, namely IMAS (Improved Maize for African Soils) and DTMA (Drought Tolerant Maize for Africa), were used for understanding the genetic architecture of MLND resistance. The objectives of the study were (1) to evaluate the diverse array of tropical and subtropical maize lines for their responses to MLND under artificial inoculation; (2) to identify genomic regions, SNPs, and putative candidate genes associated with MLND resistance; and (3) to assess the potential of GS for MLND resistance in maize.Two AM panels constituted under two major projects in Sub-Saharan Africa, namely DTMA (Drought Tolerant Maize for Africa) and IMAS (Improved Maize for African Soils), led by the Global Maize Program of the CIMMYT were used in this study. The IMAS-AM and DTMA-AM panels comprised 380 and 235 lines, respectively, representing broadly the tropical/subtropical maize genetic diversity, including germplasm derived from breeding programs targeting tolerance to drought, soil acidity, and low N, resistance to insects and pathogens , both viruses were propagated on a susceptible hybrid, H614, in separate greenhouses. Infected leaf samples collected from the field were cut into small pieces and ground in a mortar and pestle in grinding buffer . The resulting sap extract was centrifuged for 2 min at 12,000 rpm. Carborundum was added to decanted sap extract at the rate of 0.02 g/ml. The susceptible hybrid H614 at two leaves stage was inoculated by rubbing sap extract onto the leaves. Two separate, sealed greenhouses were maintained for SCMV and MCMV inoculum production. Three weeks before inoculation, ELISA was undertaken on random samples of leaves from the SCMV and MCMV greenhouses to confirm the inoculum purity.2. The presence of both viruses in the field trials was confirmed by ELISA once disease symptoms were apparent (approximately 2-week post-inoculation).In order to keep uniform MLND pressure across field trials, the optimized combination of SCMV and MCMV viruses (ratio of 4:1) were mixed and inoculated twice at 5th and 6th week after planting. Plants were inoculated using a motorized, backpack mist blower . An open nozzle (2-inche diameter) was used to deliver inoculum spray at a pressure of 10 kg/cmAll inbred lines were evaluated in one-row 3 m plots with two replications in alpha lattice design in three seasons during 2012–2014 at Narok  and Naivasha  in Kenya. All standard agronomic management practices were followed. Disease severity was scored for MLN at three-week post-inoculation. Inbreds were rated visually on a 1–5 disease severity scale, where 1 = no visible MLN symptoms, 2 = fine chlorotic streaks mostly on older leaves, 3 = chlorotic mottling throughout the plant, 4 = excessive chlorotic mottling on lower leaves and necrosis of newly emerging leaves (dead hear), and 5 = complete plant necrosis.Yijko = µ + gi + lj + rkj + bojk + eijko, where Yijko was the phenotypic performance of the ith genotype at the jth environment in the kth replication of the oth incomplete block, µ was an intercept term, gi was the genetic effect of the ith genotype, lj was the effect of the jth environment, rkj was the effect of the kth replication at the jth environment, bojk was the effect of the oth incomplete block in the kth replication at the jth environment, and eijko was the residual. Environments and replications were treated as fixed effects and the other effects as random. Heritability on an entry-mean basis was estimated from the variance components as the ratio of genotypic to phenotypic variance. In addition, best linear unbiased estimates (BLUEs) were estimated across environments assuming fixed genotype effects. For association analyses, best linear unbiased prediction (BLUP) of each line was calculated for across environments.For data based on ordinal scales, it is important to evaluate whether the data meets the assumptions of the applied statistical model  was determined by considering the pattern of the Q–Q plot of the model and the point at which the observed F test statistics deviated from the expected F test statistics . Within the local LD block including associated SNPs, the filtered genes in MaizeGDB (http://www.maizegdb.org) containing directly or adjacent to each associated SNP were considered as possible candidate genes for MLND resistance.BLUP of each line was used as phenotypes in AM scans. MLND severity data were corrected for population structure using general linear model (GLM), as well as population structure and kinship (Q + K) using mixed linear model (MLM) algorithm  was estimated. The accuracy of GS was calculated as rGS = rMP/h . For both the panels, moderate yet significant correlations were observed among the genotypic values estimated in each environment (Supplementary Table S1). This ruled out the possible bias due to environment-specific disease responses in a combined analysis. Analysis across environments revealed higher average diseases severity in DTMA-AM panel (3.53) than IMAS-AM panel (2.98) in 1–5 disease scale . These significantly associated SNPs individually explained 8–10 % of the total genotypic variance, whereas together explained 30 % of the total proportion of genotypic variance for resistance to MLND. In the DTMA-AM panel, we detected six significant marker–trait associations which individually explained 14–18 % of the total genotypic variance and together explained 37 % of the total proportion of genotypic variance for MLND resistance  the earlier studies , MCDV (Maize chlorotic dwarf virus), MSV (Maize mosaic virus), and WSMV  genes and is well characterized in several crop plants


In ecology and evolution generalized linear mixed models (GLMMs) are becoming increasingly used to test for differences in variation by treatment at multiple hierarchical levels. Yet, the specific sampling schemes that optimize the power of an experiment to detect differences in random effects by treatment/group remain unknown. In this paper we develop a blueprint for conducting power analyses for GLMMs focusing on detecting differences in variance by treatment. We present parameterization and power analyses for random-intercepts and random-slopes GLMMs because of their generality as focal parameters for most applications and because of their immediate applicability to emerging questions in the field of behavioral ecology. We focus on the extreme case of hierarchically structured binomial data, though the framework presented here generalizes easily to any error distribution model. First, we determine the optimal ratio of individuals to repeated measures within individuals that maximizes power to detect differences by treatment in among-individual variation in intercept, among-individual variation in slope, and within-individual variation in intercept. Second, we explore how power to detect differences in target variance parameters is affected by total variation. Our results indicate heterogeneity in power across ratios of individuals to repeated measures with an optimal ratio determined by both the target variance parameter and total sample size. Additionally, power to detect each variance parameter was low overall  and decreased with increasing variance in non-target random effects. With growing interest in variance as the parameter of inquiry, these power analyses provide a crucial component for designing experiments focused on detecting differences in variance. We hope to inspire novel experimental designs in ecology and evolution investigating the causes and implications of individual-level phenotypic variance, such as the adaptive significance of within-individual variation. Recent advances in computing power and access to increasingly sophisticated statistical tools such as generalized linear mixed effects models are changing research in ecology, evolution and behavior. Research questions and data analyses are no longer confined to the assumptions of clean experimental designs based on agricultural plots and Normal error distributions. Researchers now commonly incorporate multiple levels of hierarchical nesting  and can analyze data using a wide array of non-Gaussian error distribution models. This change is epitomized by the recent increase in use of linear and generalized linear mixed models . Despite the increasing use of GLMMs in ecology and evolution and growing interest in variance, we are aware of no papers that present power analyses for statistical tests on variance using GLMMs, and only one paper presenting power analyses for fixed effects in GLMMs . Here we discuss the ratio of groups to total observations within groups (n∗j), and consider how varying n and j affect power to detect each variance parameter. Second, we explore how power to detect differences in specific variance parameters is affected by increasing variation in non-target parameters . We consider both random-intercepts and random-slopes models because of their generality as focal parameters for most applications, and choose to focus on the extreme case of hierarchically structured binomial data because binary response data  contains the least possible amount of information per observation and yet is a common data format for a variety of endpoints measured in ecology.Here we present parameterization and power analyses for random-intercepts and random-slopes GLMMs that test for differences in variation among- and within-groups . We focus on three key parameters: (1) Among-group variation in intercept; (2) Within-group variation in intercept; (3) Among-group variation in slope. We examine each of these comparisons in two contexts. First, we describe the optimal ratio of groups  to observations within groups  that maximizes power to detect differences in each variance parameter. In experiments with binomially distributed response variables, observations within groups are organized into We use vocabulary and examples from behavioral ecology to illustrate our models because of their immediate applicability to emerging questions in this field. Specifically, we evaluate power to detect significant differences in among-individual variation in reaction norm intercept and slope, and within-individual variation in intercept between individuals  . Our metWhile we focus on behavioral ecology as the primary application for these power analyses, these analyses are generally appropriate for comparing variation in hierarchically structured data. For example, similar methods could be used to evaluate power to detect the effects of a new experimental district-wide policy on variation among schools in student performance, or to evaluate variation among individuals in foraging success between populations  within multiple sampling occasions (j > 1) are available for each individual. Under this sampling scheme multiple probabilities of “success”  are available for each individual, which is necessary for quantifying within-individual variation . However, we note that often in behavioral ecology only a single Bernoulli observation (n = 1) is available for each sampling occasion j. We include a description on how to modify this general case to accommodate single observations per sampling occasion in In an effort to present a framework that is customizable for a diversity of research problems, we focus on a general sampling scheme in which several Bernoulli observations  to illustrate the variance components we are interested in  and theiijky is of individual i, at level j of the covariate X  in treatment k. This model is composed of three components: the treatment mean in environment j (βk0 + βk1ijkX), the unique average response of individual i across the environmental gradient (bk0 + bk1ijkX), and a residual error due to the variation around the mean of individual X and among all individuals in treatment k, but is allowed to vary by treatment. Individuals vary from the treatment mean reaction norm in both their intercept (bik0) and slope (bik1), which together compose the total phenotypic variance attributable to among-individual variation. This individual contribution is quantified using a random intercepts and slopes model with a multivariate Normal (MVN) distribution σk01. In a LMM, the linear predictor directly predicts the mean, as shown by the identity link function in g(x), which must be linearized through the use of non-identity link functions; for example, we use the standard logit (log-odds) link for Binomial GLMM.A two treatment linear mixed model can be written as: bik0) for each individual  is drawn from a treatment-specific Normal distribution.In behavioral ecology among-individual variation in intercept ividuals . In fielment see . Previoument see . Yet, coment see . For exaWithin-individual variation in intercept sonality  or more sonality . For exasonality . LMMs casonality . Here weSubstantial empirical work has shown that individual animals in a variety of taxa display variation in phenotypic plasticity ; using mWe assess power of a binomial GLMM for detecting differences in variation by treatment. This model can be written as: ijky is the number of “successes” in ijkN observations of the ith individual in treatment k at the jth sampling occasion. When an environmental covariate (X) is present, we assume one sampling occasion occurs at each level of the covariate j. Here, in the absence of an environmental covariate, the linear predictor reduces to ijkη = β0 + bik0 + ijkv and the jth occasion is simply a repeated sampling occasion in the same conditions. Note, when ijkN = 1 there is only 1 observation per sampling occasion j, making ijky a Bernoulli response variable  and thus within-individual variation is not identifiable.Here, able see . When yiijkπ describes the underlying probability of individual i in treatment k at occasion j exhibiting a behavior. Variation in π isdetermined by the linear combination of predictors on the logit (log-odds) scale: group intercept (β0), group slope (β1), individual unique intercept (bik0), slope (bik1), and observation level overdispersion that decrease predictive power at each observation (ijkv). This linear predictor is transformed with the inverse-logit link to produce ijkπ, which follows a logit-Normal-binomial mixed distribution.In this model π between each sampling occasion j for each individual. Here the magnitude of overdispersion is allowed to vary by treatment  and Bernoulli observations within sampling occasions (n). Next, we describe simulations that evaluate how increasing “noise” (variation in non-target random effects) affects power to detect differences in targeted variance comparisons.All data were simulated in the R statistical programming environment using newly developed simulation capabilities of the For both scenarios we simulate data with biologically relevant parameter values that illustrate common trends in power. At extreme parameter values the trends presented here may not hold due to interactions between the variance components that arise at the boundaries of binomial space. We do not dwell on these exceptions since they are unrealistic for most empirical data sets, but suggest exploration of these exceptions with code provided in σ2 = 0 , we adopted the standard correction of dividing all p-values by 2  power to detect differences in the null case exceeded 5% (∼10–15%), possibly inflating power in these cases. Regardless, random effects cannot be reliably estimated with such low sample sizes and therefore in most cases such experimental designs should be avoided.We ran 2,800 simulations for each combination of parameter values. The significance of a given random effect was assessed using likelihood ratio tests (LRTs) between models with and without the focal random effect. To correct for the known conservatism of the LRT when testing for ues by 2 . This coMost researchers face limitations imposed by time, money and access to samples, and are therefore confronted with the question of how resources should be divided between individuals and measures within individuals. To investigate the optimal allocation of sampling effort between the number of individuals and number of observations per individual, we simulated two data sets for each variance comparison , we manipulated either the ratio of individuals to sampling occasions  is never suggested for a random effect, we include this combination as an illustration of the low power that results from an ill-conceived sampling scheme. For each variance comparison we simulated three different effect sizes .First, using three hypothetical total numbers of Bernoulli observations ment}σ1k2 and 3. Cent}σvk2 . For comT  from the simulations described above. For example, for comparisons of Next, we simulated data sets with increasing numbers of Bernoulli observations for comparisons of ent}σ1k2  and 5B aent}σvk2 . For theβ0 and β1 were constrained to a single value for all treatments. For comparisons of among-individual variation in intercept no environmental covariate was used causing each sampling occasion to occur in the same conditions. Additionally, In all Scenario 1 simulations, both Our goal in Scenario 1 was to isolate changes in a single variance parameter, but exploration of the dependence among multiple variance components and the mean may be warranted if it is relevant for a specific problem. Incorporating concurrent changes in intercept, slope and overdispersion parameters can be easily implemented with slight modifications to the code presented in the online supplement. We show initial results of relaxing some of these assumptions in Scenario 2, but full exploration of these possibilities are beyond the scope of this paper.T sampling scheme used in Scenario 1 and an ES of a 2.5× difference in standard deviation by treatment.Decreasing the ratio of the variance in the target random effect to total variance influences power to detect differences in the target variance among treatments. Therefore, we simulated four levels of “noise” (magnitude of non-target random effect variance) assuming a Normal distribution with increasing standard deviations  . These cT) and effect size (ES)  –5. For accasions  and 3, bccasions  and 5B. T = 600)   (At a low sample size (TSST = 600)  power toES = 2×)  power is.5×, 3×) , power iT = 300, ES = 2×), power to detect T = 600, ES = 2.5×, 3.0×; TSST = 1,200, ES = 2.5×, 3×) favor ratios heavily weighted towards having more individuals versus more repeated measures  power to detect T = 2,400, ES = 3.0) power is maximized at a ratio of individuals to Bernoulli observations of approximately 5:6  . FinallyT and relatively large effect sizes (3 SD difference between treatments), >80% power is retained across many different combinations of individuals to repeated measures for each component of variance , and differences in which sampling scheme maximizes power for different components of variance –5. As exvariance –5. Not svariance .T and ES are large . As in the two-treatment scenario, differences in variance among treatments in a multi-treatment scenario can be evaluated with a likelihood ratio test.Finally, these power analyses were created for a two-treatment scenario—“homogenous” environmental variation treatment and a “variable” environmental variation treatment. However, it is commonplace to have more than two treatments. Fortunately, our framework for conducting power analyses can be easily generalized for exploring power for experiments with more than two treatments see . In addiDespite the ubiquity of random intercepts and slopes GLMMs in ecology, evolution, and behavior, the use of GLMMs to compare variance components among populations or among experimental treatments is rare. We hope the power analyses presented here will spur novel empirical research and assist readers in constructing appropriate experimental designs and statistical models to test how variance components are shaped by ecological and evolutionary processes. We emphasize a clearly defined biological question and designation of the focal random effect when designing experiments for this application due to unique ratios of individuals to repeated measures required to optimize power for each variance parameter and low overall power . Finally, we call for future work analyzing the accuracy and precision of estimates comparing random effects by treatment for GLMMs (which our code facilitates) similar to the work of 10.7717/peerj.1226/supp-1Supplemental Information 1Click here for additional data file.


The present study aimed to predict the performance of maize hybrids and assess whether the total effects of associated markers (TEAM) method can correctly predict hybrids using cross-validation and regional trials. The training was performed in 7 locations of Southern Brazil during the 2010/11 harvest. The regional assays were conducted in 6 different South Brazilian locations during the 2011/12 harvest. In the training trial, 51 lines from different backgrounds were used to create 58 single cross hybrids. Seventy-nine microsatellite markers were used to genotype these 51 lines. In the cross-validation method the predictive accuracy ranged from 0.10 to 0.96, depending on the sample size. Furthermore, the accuracy was 0.30 when the values of hybrids that were not used in the training population (119) were predicted for the regional assays. Regarding selective loss, the TEAM method correctly predicted 50% of the hybrids selected in the regional assays. There was also loss in only 33% of cases; that is, only 33% of the materials predicted to be good in training trial were considered to be bad in regional assays. Our results show that the predictive validation of different crop conditions is possible, and the cross-validation results strikingly represented the field performance. Maize breeding programs aim to create increasingly productive hybrids, which require conducting several crosses and generating several hybrids to be tested in trial networks. Although there are various breeding strategies, which range from population breeding to the creation and selection of haploids for the synthesis of lines, the choices of parentals and cross designs  are key issues to obtain high performance hybrids.Although there are several methods available for planning crosses, the choice of method generally takes into consideration either the breeder's “experience,” desirable traits of the parental lines, and preestablished heterotic groups . AlthougAnother factor to be considered when choosing crosses is the number of lines to be used during hybrid synthesis. As the number of lines increases, the range of possible crosses can become so large that it economically precludes the establishment of all crosses . TherefoThe first studies for predicting single cross hybrids using molecular markers were conducted using the genetic distance between lines. In a naïve view, the more divergent the parentals are, the greater the heterosis is expected to be because heterosis may be described by the genetic distance and the dominance effect when disregarding the epistasis .Several studies have been conducted, starting in the 1990s, that aimed to predict the performance of hybrids based on genetic distance using this simplified theory of quantitative genetics –10. HoweNew approaches were developed in parallel to the prediction using markers that were based on studies performed by  in mixedRecently, Schrag et al. , 16, 17 The TEAM method has proved to be more efficient than the method proposed by  in prediDuring the winter of 2010, 51 lines of different backgrounds were crossed to create the test population in the experimental field of Uberlândia South of Minas Gerais State, Brazil. Fifty-eight single cross hybrids were created starting from of these lines in an incomplete partial diallel system . These m·ha−1. The tillage and cover fertilizations were conducted according to the recommendations for each area, and cultural treatments were conducted to control the populations of Fall armyworm (Spodoptera frugiperda) and moth larva (Helicoverpa zea), as well as for weed control.The two experiments from both years were conducted using incomplete blocks designs with 2 replicates, and each plot had 4 to 5 m rows, with 70 cm spacing between rows. A complete description of this experimental design can be obtained at appendix notes in Melo et al. . The graSeventy-nine microsatellite markers were used to genotype the 51 lines . These mt of marker m in line i as 1, and the absence of the allele was designated as 0. This coding scheme facilitated the design of the additive and dominance matrices of the hybrids.The data matrix of markers was designed using the presence of allele a is the genotype of the tth allele of marker m in the lines i and j, ∀ means for all situations where one or (∨) other lines present different alleles.Using the above coding scheme and considering that recombination is irrelevant in homozygous lines, the additive matrix of hybrids was designed as follows:tth alleles of marker m. A deviation of dominance for the homozygous allele occurs in the first case, and a deviation for heterozygous complement occurs in the second case.Similarly, the matrix of dominance effects was designed using the following relation ; Z1 to Z6 correspond to incidence matrices of random effects; the parameters a1, a2, d, i1, i2, and w correspond to the GCA effects of group 1 (GCA1), the GCA effects of group 2 (GCA2), the SCA, the interaction GCA1 × Environments, the interaction GCA2 × Environments, and the interaction SCA × Environments, respectively; and ξ is the model residue. Estimates of fixed effects, phenotypic variance components, and predictions of random effects were obtained by restricted maximum likelihood (REML), using the expectation-maximization (EM) algorithm.The following linear model was assumed in the first case:A corresponds to the matrix of incidence for additive effects (a) and Δ corresponds to the matrix of incidence for dominance deviations (d), both described above in molecular data topic. Unlike the RR-BLUP model, which is normally used in genomic selection, model 2 is characterized as a mixed model because the environmental effects are considered in the fixed effects matrix; furthermore, the interaction of the values assigned to each marker per environment was confounded with the residue due to high computational cost involved in estimating the interaction effects of each allele and the allelic interactions with the environment.The incidence matrices of parental effects and specific combination in the markers model were replaced by the matrices of additive and dominance effects of markers. Thus, the TEAM model was calculated as follows:i was recovered by the sum of additive effects of each allele within the specimen; that is, αi = ∑j=1nλaj : λ = {0,1, 2} and δi = ∑j=1kϕdj : ϕ = {0,1}, where k is the total number of allelic interactions l within each marker m, n is the number of alleles found in m markers, and λ and ϕ are indicator variables related to state of the marker m in the hybrid i for additive and dominance effects, respectively. It is equivalent to the point prediction αi = Αi×n)(a and δi = Δi×k)/n and σd2 = (∑j=1kdj2 + trace[InW2−1σ2])/k, where Wn−1 and Wk−1 are submatrices of the inverse matrix of the mixed model equations . The heritability was 0.65, and the phenotypic variance was strongly affected by the interaction between the hybrids and the environments. The interaction between the hybrids and the environments was 3 times greater than the genetic variance and 5 times greater than the residual variance (σiCGC2 = 0.45 t2 · ha−2  σiCEC2 = 0.23 t2 · ha−2), indicating that the performance of the hybrids was highly affected by their interactions with the environments.The genetic variance of the general combing ability of group 1 was 0.055 (tσg2 = nσa2 + kσd2 = 0.25 t·ha−1). This result can be confirmed in The analysis using molecular markers in the lines showed 636 different alleles, with a mean of 8.05 alleles per locus. The crosses performed, shown in The process of cross-validation showed that the prediction ability of the TEAM-1 and TEAM-2 methods was reasonable to low, and the model including the dominance effects was slightly superior to the strictly additive model, as shown in R2 = 0.09  and their performance in the regional assays (second-year tests) is shown in The genotypic values of SCA and GCA were broken down into 1804 additive and dominance effects, and the sum of those effects nearly recovered the genetic variance of the hybrids. The improvement in prediction using markers was not higher because the TEAM models did not include the interaction with environments, and that effect was confounded with the residual variance. The interaction between genotypes and environments was strongly involved in the phenotypic variance of hybrids and was the primary cause for the lack of genetic correlation between hybrids in the 6 locations evaluated in the second-year tests; therefore, the prediction of genotypic value might have been better if the interaction with the environment had been included.R2 (0.16–0.65) using an imbalance of 50% in a set of 400 tested hybrids and more than 1,000 molecular markers. Our results reached a fit of 0.15 for that level of imbalance using fewer than 10% of the number of markers and fewer than 15% of the number of hybrids in the test population than were used by Schrag et al. In contrast, in comparison to the study by [The prediction results found in the present study are encouraging, considering the test population size (58 hybrids), the number of markers used (79 markers), and the completely different environments between the evaluation locations of the first- and second-year tests. For example, Schrag et al. , 17 founstudy by , who obtAnother key result found in the present study was the ineffectiveness of using genetic distances to predict the performance of maize hybrids. There was only a 0.10 correlation between genetic distance and grain production, which is a nonsignificant value according to the Mantel test. Similar results were found by , 15, 24.Regarding the methods used to validate predictions, the studies published in the literature have usually used cross-validation as a method for assessing the accuracy of predictions and have not tested the predicted value of hybrids in conditions that differ completely from the test conditions; that is, under different environments and in competition with genotypes that are completely different from those used in cross-validation. Accordingly, the present study stands apart from all others by using the method efficacy to an extreme of prediction conditions and by considering that parameter for validating the method. The results found in the present study show that the accuracy of prediction for hybrid performance in locations and conditions of cultivation different from the test population is similar to that found under levels of imbalance above 30% . Thus, cFurthermore, the accuracy value was noticeably more stable than the credibility of selection assessed by using heritability because the estimated heritability was 0.65 and the realized was only 0.23. The value of 0.23 was calculated through the regression of the genotypic values of 58 hybrids tested in the training population and their genotypic values observed in the regional assays. The adjustment was 0.13, slightly above the adjustment for 175 hybrids that were not tested in the training population, which was 0.09 .Another interesting result was the ability of the method to select hybrids correctly. A function of 0 and 1 type loss is reached when analyzing The present study demonstrated that the prediction of hybrids, especially maize, still requires further study. The increase in the number of markers and individuals in the test population may be crucial for validating the method. However, it became evident that the validation in regional assays, especially in locations and conditions of cultivation different from those adopted in the test population, may provide a wider perspective on the method efficacy to the breeder. Accordingly, the results from this study suggest that validation in different conditions of cultivation is possible, and the cross-validation results strikingly represent the field performance.


Availability of affordable and accessible whole genome sequencing for biomedical applications poses a number of statistical challenges and opportunities, particularly related to the analysis of rare variants and sparseness of the data. Although efforts have been devoted to address these challenges, the performance of statistical methods for rare variants analysis still needs further consideration.We introduce a new approach that applies restricted principal component analysis with convex penalization and then selects the best predictors of a phenotype by a concave penalized regression model, while estimating the impact of each genomic region on the phenotype. Using simulated data, we show that the proposed method maintains good power for association testing while keeping the false discovery rate low under a verity of genetic architectures. Illustrative data analyses reveal encouraging result of this method in comparison with other commonly applied methods for rare variants analysis.By taking into account linkage disequilibrium and sparseness of the data, the proposed method improves power and controls the false discovery rate compared to other commonly applied methods for rare variant analyses. Decreasing costs and increasing quality have made discovery and genotyping of rare variants, which refer to variants with minor allele frequency (MAF) less than 0.05, more accessible across a large proportion of the genome and in large sample sizes. As a result of rapid expansion of human populations, there are very large numbers of rare variants segregating and these rare variants are relatively recent in origin [Despite success in detecting associations of common variants with complex traits . FurTransitioning from common variant analyses to rare variant analyses creates three challenges related to sparse data . First, m rare variants X= in a sample of n individuals having a quantitative trait y= measured on each individual. In a typical whole exome or genome sequencing scenario, m is several orders of magnitude larger than n. To combat over-determination, the typical analysis considers a subset of the variables at a time defined by physical proximity (e.g. a window) instead of functional characteristic (e.g. an annotated gene or enhancer element) because the vast majority of the rare variants are in noncoding regions in the genome. Interpretation of the results requires adjusting for multiple comparisons using accepted experiment-wise error or false discovery rate methods. Assume for the kth subset of X denoted by p<n, we haveThe CCRS method is applicable for all variants, but in this presentation, we focus on the analysis of rare variants because they pose special opportunities (i.e. large effects sizes) and challenges (i.e. sparsity). Assume we have detected and genotyped ε= is an error vector, Σ is an n×n diagonal matrix; α is the overall mean; T is an n×q covariate matrix, which includes non-genetic predictors such as age, sex and race; βk and θ are p-vector of genetic effects and q-vector of non-genetic effects, respectively. Although model \docr is the rank of X, which is smaller than min; ∥.∥a denotes La norm; and X such that d1≥d2≥…≥dr; vj and uj are the jth columns of V and U respectively. The L1 norm penalization is equivalent to vij is ijth entry of V, provides sparse principal components, UD. This is an optimization problem equivalent to maximizing uj and vj under constraint .In this context, it is not appropriate to select only the first few principal components as is usual in many applications, but rather we select the PCs based on their degree of association with the phenotype. To simultaneously measure the genotype-phenotype association and carry out variable selection, we consider a linear regression model including a concave penalization with loss functionZ=UD indicates a matrix of computed PCs with corresponding effect size γ, κ∈ and regularization parameter where γ, in the sense that some of components of γκ,ν) and prog., [κ=1 .dbGAP (http://www.ncbi.nlm.nih.gov). Each region includes 50 variants  sequenced for 1456 individuals. Based on our experience, we have found that 50 variants are appropriate to capture the LD structure. As an example, Fig. To evaluate the performance of the CCRS method, we randomly identified 1000 regions from a real whole genome sequence data set available from phs000668 study in % of variants in each influential region to be causal variants with effect size +1 for Model-1 and Model-3 and with effect size ±1 for Model-2 and Model-4. In Model-5 and Model-6, the number of causal variants in a region is increased to 20 % of the total variants with different effect sizes randomly selected from U and {U,U}, respectively, where U denotes uniform distribution. Hence, we considered models with the same and also different effect directions.We considered six different phenotypic effect scenarios Table . We firsModel-1 and Model-2, the causal variants are correlated with some neutral variants in their regions but in Model-3, Model-4 they are uncorrelated. For Model-5 and Model-6, both correlated and uncorrelated variants are selected (10 % of each).To obtain a better understanding about the effect of LD on the result of the analysis, we selected variants based on their correlations. In In rare variants analysis, we are interested in identifying regions with significant effects on the phenotype corresponding to the following set of hypotheses for each regionTo test these hypotheses, we calculated the likelihood ratio of the selected model based on CCRS to the Null model, which does not include genotype variants in the model.βk in I+ρ11T where I is an p×p compound symmetric matrix, and 1T=T.We evaluated the performance of the CCRS method compared to four other commonly applied methods: Collapsing  denoted mentclass2pt{minimα=0.01 and 0.05 and calculated false discovery rate (FDR) and true positive discovery rate (TPR) defined asTo examine the impact of significance level on the false and true discovery rates, we considered both F is the number of false positives; T is the number of true discoveries; R is the total number of significant regions; and M is the total number of regions.where ν = 0.01 and κ = 0.5 after calculating BIC of the model over for different values of ν in {0.001,0.005,0.01,0.02,0.05}. Here, BIC of the model is defined asd(ν) is the number of effective parameters, ν [ν might be applied for problems with larger number of variants in each region.To select the best model based on the CCRS method, we set  given ν . A largeModel-1 and Model-2 are shown in Fig. α=0.05, when the direction of effects are the same. At the level α=0.05, the CCRS method shows better performance than SKAT-O when the direction of effects are different. At the level α=0.01, the CCRS and SKAT-O show nearly the same performance. It is clear from the figure that the CCRS method improves performance over the SPC method.The results of the simulation study for Model-3 and Model-4. The CAST method for Model-3 and the Col method for both Model-3 and Model-4 show poor performance. In both models, the CCRS shows noticeably better performance in both α levels. The influential regions in Model-1 through Model-4 have the same effect sizes on the phenotype. Hence, comparing Figs. Figure Model-5 and Model-6 which include both correlated and uncorrelated effective variants. SKAT-O shows smaller FDR at the level α=0.05 in the left panel and slightly smaller TPR than CCRS, although at level α=0.01 CCRS shows better performance in terms of FDR and TPR. When the directions of effects are different (Model-6), right panel, CCRS outperforms the other methods.Figure The result of this simulation study shows that the CCRS performs better and more robust than other methods under a variety of genetic architectures, and it is much more prominent when the causal variants are not correlated with neutral variants in the region. Neglecting the presence of LD leads to overestimation of the overall effect of the regions. Although this overestimation might increase the power of detecting a region with some small effects that are correlated with some neutral variants, it increases the risk of missing more promising regions in procedure of multiple comparison of hypotheses testing.We analyzed sequencing data from the Atherosclerosis Risk in Communities (ARIC) study . The datThe genotype data includes 949,986 rare variants that are mostly in noncoding regions in the genome . Therefo−6, we detected one significantly associated region by the CCRS method. Figure p-values of 80 windows around this region. All of the approaches except CAST and Collapsing test have a peak in this region. The figure shows that the CCRS maintains power for detecting phenotype-influencing region while keeping the p-value of the null or neutral regions small. This is an important property of CCRS that controls the false discovery rate.To define the threshold for statistical significance taking into account multiple hypothesis testing, we ran 100,000 permutation test over 100 windows. Based on this threshold, 10http://www.sabiosciences.com and [The region contains the gene, FAM78B, which is expressed at high levels in myocytes, fibroblasts, endothelial cells. Little is known about the function of FAM78B. However, within the promoter for FAM78B, three binding sites for the transcription factor PPARG and two binding sites for the transcription factor HNF1A have been identified (.com and ). Pi et .com and  have sho.com and . PPARs a.com and ).We have introduced a new approach, CCRS, for the analysis of whole genome sequence data in order to identify regions of the genome  influencing a phenotype of interest. The CCRS improves the power of identifying a set of variants associated with a phenotype by taking into account the sparseness and LD structure in the data. The CCRS applies a concave penalized regression method after projecting the sequence variants in a full rank space that is more informative via sparse principal component analysis. By applying sparse PCA, the CCRS aims to enhance the information in the predictors instead of reducing dimension as typical application of sparse PCA, which might increase risk of missing important variants in rare variants analysis. While the first step of analysis (sparse PCA) is an unsupervised method, it does not increase the FDR of the method in the second step of the analysis.Although the CCRS method can be applied to both common and rare variants, the focus of this analysis was on rare variants because of the role of these variants on phenotype variation. The CCRS method also can be easily expanded to logistic regression and applied for case control studies. However, we investigated the CCRS performance for quantitative traits while the overwhelming majority of the literature focusing on case/control studies and there is a daunting need to develop methods for quantitative traits.Using simulated data, we show that the FDR of the CCRS method is smaller than other commonly applied genomic region-based test methods while it has higher power of identification in most of the situations. Furthermore, the FDR of CCRS is smaller and robust to the LD structure in the region in comparison to the other methods. While the statistical test for rare variants are typically region-based test, there is risk of overestimation of overall effect of regions by neglecting the LD between causal and neutral variants in the region. Consequently, the risk of missing promising regions might increase through multiple hypotheses testing.L1 norm constraints to provide a full rank design matrix. Imposing L1 penalization in PCA generates a sparse loading matrix that renders the analysis interpretable. The CCRS method efficiently incorporates information from low frequency variants by generating new predictors that are much more informative. The CCRS uses a concave penalized regression model to simultaneously select the most important PCs regarding their association with the phenotype of interest, but also to estimate their effect sizes. The zero effect sizes can be uniquely identified due to the use of full rank approximation of the design matrix. The advantage of the concave penalty term is that the rate of shrinkage gets smaller as the effect size increases. In other words, the CCRS not only has the property of parsimony, it also avoids shrinkage over large effect sizes. Thus, the CCRS maintains power for detecting phenotype-influencing regions while keeping the p-value of the neutral regions small.Penalized regression and other shrinkage methods that have been introduced for sparse data applications can correctly select nonzero coefficients under specific conditions , 44. Apphttp://www.proteinatlas.org/ENSG00000188859-FAM78B/tissue). In a real application, annotation of the non-coding regions should be integrated into the analysis, and replication in an independent sample would be the next step to consider it as a novel discovery.As an example real data application, we used the CCRS method and genome sequence data to analyze plasma ApoA1 levels, and one region met the experiment-wise criterion for statistical significance. The region contains the gene, FAM78B, which is expressed at high levels in myocytes, fibroblasts, endothelial cells (Large-scale whole genome sequencing and high-powered computing are becoming more readily available and affordable. There is an emerging shift from sequencing and computing technologies toward study design, data processing algorithms, and statistical and informatics methods for extracting usable information from the very large amount of genome sequence data that are imminent. The CCRS method presented here for the first time is a practical, powerful and efficient method for taking into account the nature of whole genome sequence variation to identify regions of the genome influencing common complex risk factor phenotypes and diseases.


The broad sense heritability was 0.84 across all years. Six accessions displayed stable AE >80% across all the years. Genotyping data included 2,575 DArT markers . AE was influenced both by genotype and by the growing environment. In all, 131 significant marker trait associations (MTAs) (|log10 (P)| >FDR) were established for AE. AE behaved as a quantitative trait, with five consistently significant markers  contributing a minor to modest proportion (4.29% to 8.61%) of the phenotypic variance and affecting the trait either positively or negatively. For this reason, there is potential for breeding for improved AE by gene pyramiding. The consistently significant markers linked to AE could be helpful for marker assisted selection to transfer AE to high yielding varieties allowing to promote the exploitation of hybrid-heterosis in the key crop wheat.In a number of crop species hybrids are able to outperform line varieties. The anthers of the autogamous bread wheat plant are normally extruded post anthesis, a trait which is unfavourable for the production of F1 hybrid grain. Higher anther extrusion (AE) promotes cross fertilization for more efficient hybrid seed production. Therefore, this study aimed at the genetic dissection of AE by genome wide association studies (GWAS) and determination of the main effect QTL. We applied GWAS approach to identify DArT markers potentially linked to AE to unfold its genetic basis in a panel of spring wheat accessions. Phenotypic data were collected for three years and best linear unbiased estimate (BLUE) values were calculated across all years. The extent of the AE correlation between growing years and BLUE values ranged from r = +0.56 (2013  Thanks to the phenomenon of heterosis, F1 hybrid cultivars in a number of crop species are able to out-perform line varieties, both with respect to economic yield and to yield stability . AlthougAE like the majority of agronomically important traits, is typically under polygenic control ,10. The http://www.ipk-gatersleben.de/en/genebank/), comprised 57 accessions bred in Europe, 31 in Asia, ten in each of North and South America and one in Australia; two accessions were of unknown origin . The broad sense heritability of AE was calculated from the expression σ2g / (σ2g + σ2g.E/ nE +σ2e / nE.nr), where σ2g represented the genetic variance, σ2e the environmental variance, nE is the number of environments and nr the number of replicates.The mean AE in each growing year was used to calculate the best linear unbiased estimate (BLUE) values, assuming fixed genotypic effects. The calculation was performed by routines implemented in GenStat v16 software  applying the “Mixed models REML” module and the “Linear mixed models” function. A Pearson correlation coefficient among the three growing years was also calculated. The repeatability among the biological replicates was calculated from the expression www.triticarte.com.au); the initial number of scored dominant loci was 2,836. An allele frequency minimum threshold of 5% was applied prior to the determination of marker-AE associations which reduced the size of the data matrix to 2,575 x 111. The quality of these markers was high, with an average call rate, reproducibility and polymorphism information content value as 96.37%, 99.53% and 0.36, respectively.DArT profiling was performed by Triticarte Pty. Ltd. . The first two principal coordinates were graphed in two-dimensional space to show the clustering of different accessions.Genetic relationships amongst the spring panel accessions were also investigated graphically via principal coordinates analysis (PCoA) based on DArT genotype with the software package NTSYSpc (The Loiselle kinship matrix derived from the full set of DArT markers (MAF >5%) was generated using the SPAGeDi-1.3d program .~ N , also the residual error was assumed to be normally distributed with ~ N . The threshold of |log10 (P)| ≥2.0 was set in GenStat v16 for the detection of significant markers. However, this set of DArT markers was subjected to another test of robustness using the false discovery rate (FDR) at 0.05 [10 (P)| >FDR in different years and BLUE values individually were considered as significant markers. The markers which appeared significant (|log10 (P)| >FDR) across at least two years and BLUE values, were termed as consistently significant markers. |log10 (P)| value distributions were displayed in the form of Q-Q plots for each growing year and across years . For the set of markers unlinked to an AE QTL, the |log10 (P)| values would be expected to be uniformly distributed, and large deviations are indicative of spurious associations [Marker trait associations (MTAs) involving AE and the DArT markers were identified using the “QTL analysis” module and the “Single trait association analysis” function implemented in GenStat v16, applying a Loiselle kinship matrix (K) for the purpose of correction for population stratification. Thus each AE value was modelled by the expression μ + marker effect + genotype effect + residual error, where the genotype effect was assumed to be normally distributed with  at 0.05 . The det at 0.05 . Thus, tciations ,28.2). The additive contribution (effect) of each linked marker was calculated with GenStat v16: negative effects reflect a decrease in AE , and positive ones an increase . A partial linkage map was constructed with the significant markers (|log10 (P)| >FDR) having known map location and non-significant markers ensuring even genome coverage ; this wg et al. .Using the set of mapped DArT markers genome-wide linkage disequilibrium (LD) was studied in the whole panel of spring wheat accessions. Intra and inter-chromosomal quantification and graphical representation of LD heat maps and intra-chromosomal LD decay was accomplished with GenStat v16.i.e., direct scoring of the anthers in the field as in the years 2013 and 2014 and collecting and freezing the spikes and determining the AE in the lab. The two methods used for scoring in 2015 were strongly correlated with one another . The laboratory-based method used in 2015 was preferred for the subsequent genetic analysis, largely because it avoided problems associated with the lodging of the many non-semi-dwarf accessions. AE scores varied markedly between the accessions. The years 2013, 2014 and BLUE values were close to a normal distribution, however some skewing was observed in the year 2015 which may be due to environmental conditions (vs 2015) to 0.91  (Anther extrusion (AE) was scored for 111 accessions and showed a wide range of variation among the genotypes and the years. Each of the 111 accessions according to the country of origin and their AE scores are given in nditions . The minnditions . The obsnditions  which co values) . The lev values) , and the values) .The imposition of the minor allele frequency (MAF) threshold >5% reduced the DArT marker set by 261. Of the remaining 2,575 markers, 2,166 had a known map location, which spread unevenly across all three constituent genomes; however, each of the 21 chromosomes was represented. The B-genome had the highest coverage (43.6% of the markers), while 37.7% mapped to the A genome and 18.7% to the D genome; 15.2% of the markers mapped to one of the three homoeologous group 1 chromosomes, while only 6.3% mapped to a homoeologous group 4 chromosome. The single most populated chromosome was 3B (205 loci) and the least populated was chromosome 4D (eight loci); the mean number of loci per chromosome was 103.1 .r2 declining below 0.2) extended for ca. 2.5 cM. As the least number of markers was mapped on the D-genome, for some D-genome chromosomes no LD could be calculated. Intra-chromosomal locus pairs had a higher mean r2 value (0.05) than inter-chromosomal locus pairs (0.01)  .Our Structure results using the DArT markers with MAF >5% indicated the ΔK peak at K = 2, providing support for the existence of two genetically distinct sub-groups in our association panel. Structure results are grouped and graphed with CLUMPAK . Both StA PCoA-based test for the existence of clustering among accessions is illustrated in Ppd-D1) [Vrn-1 ) [Rht-1 (Rht-B1 and -D1)) [Ppd-D1 was carried by 16 accessions. The recessive allele at the three Vrn-1 homoeoloci were present in, respectively, 75, 37 and 98 of the cultivars; although the expectation is that a spring type should carry the dominant allele at least at one of the three loci, six cultivars carried the recessive allele at all three loci (Rht-1 loci was low (three accessions were the mutant genotype Rht-B1b and one was Rht-D1b). The accession TRI 10296 was a double dwarf (Rht-B1b / Rht-D1b), and its mean plant height across the three growing seasons was only 67.0 cm. We did the MTA analysis based on candidate gene genotypes and found no significant MTA for these major genes.The allelic status of the genes underlying photoperiod sensitivity (Ppd-D1) , vernalind -D1))  and semind -D1))  was obtaree loci , a genetvs expected |log10 (P)| values, indicated the most stringent analysis for the year 2014 among all years and their BLUE values (10 (P)| >FDR), involving 99 DArT markers were identified across the three years and BLUE values | >3.0). In total, 42 of the 49 MTAs (85.71%) noted in the field-based method for AE measurement also appeared in the lab-based method | >FDR), a favourable effect (increasing effect on AE) was associated with 72 of the significant markers (2 = 0.172 (P <0.001) (2 = 0.395 (P <0.001)  , while f <0.001) . This meOf the 31 significant markers based on BLUE values , 17 had 2 = 4.29–8.61%). A similar conclusion has been reached following a conventional QTL analysis [Anther Extrusion (AE) behaved as a quantitative trait among the germplasm panel, with each consistently significant marker  contributing a modest proportion of phenotypic variance  may be significantly affected by the extent of LD across chromosomes . ReducedwPt-4131 present at 37.9 cM position and explained 8.61% of the mean phenotypic variation. The same chromosome has been implicated as the site for an AE-QTL, although not in the same region [tPt-3719 and the other with wPt-1548. Although Lu et al. [wPt-732263 acted to reduce AE. An important QTL for AE in a doubled haploid population was detected by Skinnes et al. [wPt-732263. The consistently significant marker wPt-741749 was unmapped in our data. In the literature, it was located onto the consensus map of durum wheat (Meridiano X Claudio) on chromosome 1B at 12.2 cM position by Maccaferri et al [wPt-741749. Recent literature reported on anther extrusion/retention relies mainly on recombinant inbred lines and doubled haploid lines for both spring [Significant AE-MTAs were identified on six of the seven A genome chromosomes (not 7A), all B-genome chromosomes and on three of the D-genome chromosomes . The continuous distribution of AE in our study and previous results suggest that several factors are involved in the inheritance of this trait ,10,35,36ri et al . SkinnesThe association mapping analysis has confirmed that anther extrusion (AE) in wheat is under the control of a number of gene loci, some of which act favourably and others unfavourably. As a result, there is potential for genetic advance for AE, by selecting for the former and against the latter. Thus, a breeding strategy based on marker assisted selection would need to incorporate methods of genomic selection ,49, whicS1 Fig(PDF)Click here for additional data file.S2 Figunm stands for unmapped markers.(PDF)Click here for additional data file.S3 Figr2 between markers. Intra-chromosomal plots of LD decay show r2 against the genetic distances (cM) between pairs of DArT loci in the spring wheat genome.Heat maps depict (PDF)Click here for additional data file.S4 Fig–(A). Structure results displaying continent-wise distribution of whole spring wheat panel based on DArT genotype as K = 2. The white lines mark the separation between the continents. (B) Structure harvester results displaying the plots of the (i) ΔK and (ii) log-likelihood values for the analysis based on DArT markers. K = 2 is the most appropriate.(PDF)Click here for additional data file.S5 Fig(PDF)Click here for additional data file.S6 Fig–(A). The number of (i) positive and (ii) negative AE alleles harboured by individual cultivars. (B) Linear regression of BLUE values for AE against the number of (i) favourable and (ii) unfavourable AE alleles harboured by individual cultivars. Higher numbers of favourable alleles are associated with an increased BLUE value for AE, and higher numbers of unfavourable alleles with a decreased BLUE value for AE.(PDF)Click here for additional data file.S1 Table(XLSX)Click here for additional data file.S2 TableThe AE performance in each growing year and the respective BLUE values are also given.(XLSX)Click here for additional data file.S3 Table(XLSX)Click here for additional data file.S4 TableColumns G to L record the additive effects and the R2 for each year; columns M and N indicate the additive effects and the R2 of the BLUE values.(XLSX)Click here for additional data file.S5 TableThose restricted to a single year or BLUE values numbered 73, those to one year and BLUE values numbered 42, those to two years and BLUE values numbered 12, and those to all three years and BLUE values numbered four.(XLSX)Click here for additional data file.S6 Table(A). List of MTAs appeared in 2015 field based measurement of AE. (B). List of MTAs appeared in 2015 Lab based measurement of AE. (C). Comparison between two methods.(XLSX)Click here for additional data file.S7 Table–(A). Number of favourable alleles per cultivar. A total of 72 marker alleles were associated with a favourable additive effect on AE. (B) Number of unfavourable alleles per cultivar. A total of 27 marker alleles were associated with an unfavourable additive effect on AE.(XLSX)Click here for additional data file.


Current family-based association tests for sequencing data were mainly developed for identifying rare variants associated with a complex disease. As the disease can be influenced by the joint effects of common and rare variants, common variants with modest effects may not be identified by the methods focusing on rare variants. Moreover, variants can have risk, neutral, or protective effects. Association tests that can effectively select groups of common and rare variants that are likely to be causal and consider the directions of effects have become important. We developed the Ordered Subset - Variable Threshold - Pedigree Disequilibrium Test (OVPDT), a combination of three algorithms, for association analysis in family sequencing data. The ordered subset algorithm is used to select a subset of common variants based on their relative risks, calculated using only parental mating types. The variable threshold algorithm is used to search for an optimal allele frequency threshold such that rare variants below the threshold are more likely to be causal. The PDT statistics from both rare and common variants selected by the two algorithms are combined as the OVPDT statistic. A permutation procedure is used in OVPDT to calculate the p-value. We used simulations to demonstrate that OVPDT has the correct type I error rates under different scenarios and compared the power of OVPDT with two other family-based association tests. The results suggested that OVPDT can have more power than the other tests if both common and rare variants have effects on the disease in a region. Genome-wide association studies (GWAS) have been successful in identifying common variants associated with complex diseases. However, most identified variants explain only a small portion of heritability for the complex trait Association tests for rare variants have been developed rapidly for case-control studies. The cohort allelic sums test (CAST) was the first test developed specifically for rare variant association analysis Several family-based association tests have been developed recently for rare variants. Most of them were extended from methods developed for case-control design. For example, the weighted-sum approach was applied to the FBAT multimarker test In this work, we propose to use the ordered subset algorithm T =   – . For a discordant sib pair, S =   – . Then the PDT statistic iD for a family i is defined as i. Assume that we have N nuclear families. The single-variant PDT statistic X is We first introduce the PDT statistic as it is used as the fundamental test statistic in the proposed test. An informative nuclear family for the PDT is a family that has at least one affected child and two genotyped parents, where at least one parent is heterozygous. A discordant sibship is informative for the PDT if there is at least one affected and one unaffected sibling with different genotypes. Here we consider families that contain an informative nuclear family and/or an informative sibship. Assume that allele 1 is the minor allele and allele 2 is the other allele at a variant. For a triad (two parents and one affected offspring) in a nuclear family, x at a variant, where x can have values of 0, 1, or 2 based on the minor allele count at the variant. Then the GRRs x = i and the other has genotype x = j) conditional on the fact that their child is affected. The probability We use genotypic relative risks (GRRs) to select common variants in the proposed test. We modified the approach in lr in k trio families (two parents and one affected offspring). Although we focus our discussions on trios, the method can be generalized to general nuclear family structures. In a sequencing region, assume that there are n common variants with minor allele frequency (MAF)>0.05 and tm rare variants with MAF<t for a given allele frequency threshold t. The MAF for each variant is estimated from the parental genotypes in the sample. The region can be defined as any set of variants . The PDT statistic X is calculated for each of the common and rare variants.We assume that there are n variants. The log of relative risk (lr) with respect to minor allele is calculated for each of the n variants. The absolute values of lr are ordered from large to small, and the PDT statistics are ordered based on the rank of their corresponding lr values. We use the absolute values of lr for ordering so that variants with protective effects can also be in the top rank. Assume that the ordered PDT statistics aren variants are added to the subset, the minimum p-value is calculated as follows:i variants are independent. The approach in (4) is similar to the adaptive rank truncated product (ARTP) method C as the statistic corresponding to the minimum p-value if the minimum p-value is less than or equal to 0.05. C is defined as 0 if the minimum p-value is larger than 0.05.We use the ordered subset algorithm to select a set of common variants from the t. For a set of tm variants with MAF<t, we calculate the rare variant statistic as follows:We found that the ordered subset algorithm incorporating GRRs is only suitable for selecting common variants. Calculating the ratios in iX is the single-variant PDT statistic for variant i. The combined statistic for rare and common variants at the threshold t is defined as follows:The OVPDT algorithm is summarized as follows:lr values for variants with MAF>0.05.Calculate the normalized Calculate the single-variant PDT statistics for all variants.Calculate the combined statisticsK times. For a permutation i, we obtainPermute the transmitted/nontransmitted alleles from parents to siblings simultaneously within each family. Repeat steps 2 and 4 for Standardize i.Define the OVPDT statistic iM>M)+1)/(K+1).The p-value is calculated as ((# of C) across different t. Therefore, for rare variants, only variants with MAF below a certain threshold contribute to the statistic M. This is based on the observations in the simulation studies in We compute the rare variants statistic  and SKAT (FB-SKAT) tests We show the type I error rates for OVPDT under the 10 scenarios in The power comparison for OVPDT with FB-Burden and FB-SKAT is shown in We also show the power for FB-SKAT_0.05 in http://www.broadinstitute.org/gsea). We extracted the gene sets that each pair of the gene sets do not overlap for more than 5% of the genes. A total of 442 gene sets were tested. The QQ plot for the analysis results is shown in We applied OVPDT to the Genomic Origins and Admixture in Latinos (GOAL) study dataset. The dataset consists of 25 trios genotyped on Illumina Human Exome Beadchips. Variants with missing rates>10% were removed. Variants with HWE test p-values <0.001 were also removed. The overall genotyping rate was 99.97% and there were 55,569 polymorphic variants in the data. The 25 trios do not have disease status as the aim of the GOAL project is to investigate the haplotype structures in Hispanics. Therefore, we assumed the child was affected (but affection status was unrelated to genotypes) and parents were unaffected in each trio. Although the affection status was artificially determined, the analysis results can be used to evaluate the validity of the test in real data. Because most of the genes contain only one or a few variants, we performed pathway-based analysis. We downloaded the entire gene sets from the Molecular Signatures Database (MSigDB) from the GSEA website  in C++ are used to parallelize the code. The program can finish the analyses based on 2,000 permutations for the simulated 10 kb and 25 kb regions in 1,000 nuclear families in 8.6 and 17.3 seconds, respectively, on a Linux server with Xeon 2.0 GHz CPUs with 8 threads. Therefore, the program can perform genome-wide gene-based analysis in a reasonable time frame. The program can be downloaded for free at Figure S1QQ plots for the type I error simulations.(TIFF)Click here for additional data file.