There is a great interest in understanding the genetic architecture of complex traits in natural populations. Genome-wide association studies (GWAS) are becoming routine in human, animal and plant genetics to understand the connection between naturally occurring genotypic and phenotypic variation. Coalescent simulations are commonly used in population genetics to simulate genotypes under different parameters and demographic models.phenosim, a software to add a phenotype to genotypes generated in time-efficient coalescent simulations. Both qualitative and quantitative phenotypes can be generated and it is possible to partition phenotypic variation between additive effects and epistatic interactions between causal variants. The output formats of phenosim are directly usable as input for different GWAS tools. The applicability of phenosim is shown by simulating a genome-wide association study in Arabidopsis thaliana.Here, we present phenosim to add phenotypes, the data sets can be used to assess the influence of various factors such as demography, genetic architecture or selection on the statistical power of association methods to detect causal genetic variants under a wide variety of population genetic scenarios. phenosim is freely available from the authors' website http://evoplant.uni-hohenheim.deBy using the coalescent approach to generate genotypes and  In recent years, genome-wide association studies (GWAS) became widely used to uncover the genetic basis of complex traits by comparing patterns of genetic and phenotypic variation -3. The pCoalescent simulations are widely used to simulate genotypes under complex demographies  with recphenosim, a tool written in Python [Here, we present n Python  that wasphenosim is shown in Figure phenosim is able to read the output of the ms[msHOT[msms[GENOME[The general work flow of of the ms, msHOT[6sHOT[msms and GENOms[GENOME programsFor qualitative traits, one- and two-locus models are supported. The user defines the model by setting the penetrance (probability of being affected) for all genotypes. In the two-locus model, this is done by a penetrance table for all possible allelic combinations among the two loci. Therefore, the user may define arbitrary interactions between all alleles of the loci. The case/control-status of all simulated individuals is then assigned according to the model. In many cases, disease states are caused by risk alleles segregating at low allele frequencies in the overall population. As such low frequency variants share a genealogy that may differ from high frequency variants and thus the linkage pattern around these variants may be different , the useN. We provide two different models, depending on the ploidy of the individuals. The effect of the j-th QTN is j πand the QTN has a derived allele frequency of jf. It should be noted that the sum of all QTN effects, ∑j πj, equals the heritability, h2, of the trait. If the individuals are haploid, the allelic state of the i-th individual at the j-th QTN is ija, where ij a:= 0 if the allele is ancestral and ij a:= 1 if it is derived. Then the phenotype Y of individual i is calculated as:For quantitative traits, multiple QTNs with additive effects or epistatic interactions between two QTNs are possible. By default locations of causal variants are selected randomly. Nevertheless, the user can determine the position of a QTN manually and/or restrict the selection to an allele frequency range. A phenotype is generated based on the formulas of , which wThe phenotype of diploid individuals under an additive model without dominance is calculated as:ij Q:= 1 if the j-th QTN is homozygous derived, ij Q:= 0 if the QTN is heterozygous and ij Q:= -1 if the QTN is homozygous ancestral. Dominant effects at each QTN and additive effects between loci are also supported for diploids. In this case, equation (1) is used with ij a:= 0 for homozygous ancestral QTNs and ij a:= 1 for heterozygous and homozygous derived individuals.where E πbetween these QTNs can be simulated. This epistasis is modeled as a fictive third QTN, whose allelic state iE ais 1, if the individual carries at least one derived allele at both basal QTNs. For users with a some Python scripting experience, other types of epistasis can easily be simulated by modifying the code of phenosim. To simulate a causal haplotype or allelic heterogeneity among two causal variants within a single gene, both QTNs may also be located on a common haploblock defined by the four-gamete test [If exactly two QTNs are selected, a positive, additive epistatic effect ete test .quantiNemo[quantiNemo utilizes time-consuming forward simulations, whereas phenosim allows to include epistasis between QTNs within a time-efficient coalescent framework.To our knowledge, uantiNemo is the oBlossoc/QBlossoc[EMMA/EMMAX[PLINK[QTDT/MERLIN[TASSEL 3.0 [phenosim is available as Additional File http://evoplant.uni-hohenheim.de After phenotypes have been generated, a predefined number of markers and/or individuals can be sub sampled from the total simulated population. The causal marker(s) can be optionally removed from the sample, since frequently the causal mutation itself is not genotyped in a genome-wide study. Finally, genotypes and phenotypes are written into different output file formats that can be directly used as input for commonly used association programs such as /QBlossoc,18, EMMAMMA/EMMAX,20, PLINMAX[PLINK, QTDT/MEDT/MERLIN,23 and TSSEL 3.0 . A snapsphenosim to simulate data for GWAS, we utilized GENOME[e N= 1000, with a population recombination parameter of ρ = 8 · 10-3 and 250,000 SNPs distributed over a 120 Mbp genome. These settings are comparable to data sets used for recent GWAS in A. thaliana [phenosim was then used to generate phenotypes under three different models: (i) 2 QTNs, each with an effect of 0.05; (ii) 2 QTNs at random positions, each with an effect of 0.01, and epistatic interaction of E π= 0.08; and (iii) 2 QTNs, located on a common haploblock, each with an effect of 0.01 and epistatic interaction of E π= 0.08. In all three scenarios, the total proportion of variance explained by these QTNs and their interaction was identical (h2 = 0.1). Four hundred chromosomes were sub-sampled and the causal polymorphisms were removed from the data. EMMAX[EMMAX are able to detect QTNs with small additive effects and a strong positive epistatic interaction. However, in certain situations larger samples than simulated sizes are needed and some results may be confounded by false positives as discussed earlier [To demonstrate the ability of ed GENOME and simuthaliana . phenosita. EMMAX. The hig earlier .GENOME[phenosim on a single core of an Intel Xeon X5650 (2.66 GHz) Processor. To compare this running time with other software tools, we simulated two QTNs and 249,998 neutral loci in a population of 500 diploid individuals using quantiNemo[quantiNemo generated ~120 generations. As the expected coalescent time for a sample is ~ 4e Ngenerations [GENOME in the same time. Although forward simulations like quantiNemo allow more complex demographic, selection and trait scenarios, the combination of coalescent simulators and phenosim is much more suitable for generating multiple simulations of large sample sizes.On average, a single simulation ran 4 min with GENOME and 2 miuantiNemo. In six erations , this isphenosim, one can investigate the statistical power and other characteristics of GWAS methods efficiently. Additionally, as different causal markers may contribute different effects to a trait, the essential sample size and number of markers to detect a certain pattern can be estimated.Demographic effects, genetic architecture, selection, and different mutation and recombination rates affect the ability to detect the genetic basis of complex traits in natural populations . Such poThe authors declare that they have no competing interests.Project name: phenosim• Project home page: http://evoplant.uni-hohenheim.de• Operating system(s): Platform independent• Programming language: Python• Other requirements: Python 2.X• License: no license required• Any restrictions to use by non-academics: none• TG, IG and KJS conceived the project. TG and IG designed the software. TG wrote the code. IG analyzed the data. KJS supervised the project. All authors contributed to writing of the manuscript. All authors read and approved the final manuscript.phenosim v0.15. The archive includes the current version of phenosim as well as a documentation of its usage. For updated versions, please visit the authors' website http://evoplant.uni-hohenheim.de.Click here for file


Intersexual selection has been proposed as an important force in shaping a number of morphological traits that differ between human populations and/or between the sexes. Important to these accounts is the source of mate preferences for such traits, but this has not been investigated. In a large sample of twins, we assess forced-choice, dichotomous mate preferences for height, skin colour, hair colour and length, chest hair, facial hair, and breast size. Across the traits, identical twins reported more similar preferences than nonidentical twins, suggesting genetic effects. However, the relative magnitude of estimated genetic and environmental effects differed greatly and significantly between different trait preferences, with heritability estimates ranging from zero to 57%. Expanding on ideas first proposed by Darwin The source of the preferences for these traits is important to sexual selection explanations. Darwin noted that it is “possible that certain tastes may in the course of time become inherited, though there is no evidence in favour of this belief”, and that if true it would allow for sexual selection to favour varying features in populations that have inherited different “innate ideal standard[s] of beauty” However, the source of mate preferences for human morphological traits has not been established. Sexual imprinting  appears important in a number of species Here we use a large twin sample to investigate genetic and environmental influences on forced-choice, dichotomous preferences for height, skin colour, hair colour and length, chest hair, facial hair, and breast size. It should be noted that while demonstrating significant heritability would indicate a genetic basis to these preferences, lack of significant heritability would not necessarily indicate the lack of a genetic basis, since a genetic basis could be invariant in the population, which would not contribute to heritability The study was approved by the St. Thomas’ Hospital research ethics committee. All study participants involved in this study provided informed written consent.Data were from 4,586 twin individuals aged 19 to 83 from the UK Adult Twin Registry, a cohort of unselected volunteer Caucasian twins. The sample consists mostly of females because initial research focused on diseases with higher prevalence in women than in men. There were 4,044 females (mean age = 51.0±12.7) and 541 males (mean age = 49.2±14.0), including 1,762 full pairs  and 1,060 single twins. There were too few (16) opposite sex pairs for stable correlation estimates, so these individuals were treated as single twins. Further details on the data collection and zygosity determination, and on the comparability of the twins to age-matched singleton populations can be found elsewhere Participants reported their preferred features in a partner with dichotomous response options see . Between2). By testing the change in chi-square (Δχ2) against the change in degrees of freedom (Δdf), we can test whether dropping or equating specific model parameters significantly worsens the model fit, and can thus test hypotheses regarding those parameters.In accordance with standard genetic analysis of twin data, all analyses employed maximum-likelihood modelling procedures using the statistical package Mx Variance in the preferences was partitioned into that due to additive genetic (A), nonadditive genetic (D), shared (family) environmental (C), and residual influences (E). This can be achieved because MZ twins share all of their genes, while DZ twins share on average only half of their segregating genes. A, D, C, and E influences predict different patterns of MZ and DZ twin correlations, and structural equation modelling is used to determine the combination of influences that best matches the observed data. A limitation of the classical twin model is that there is not enough information to estimate C and D simultaneously; C is estimated if the DZ twin correlation is more than half the MZ twin correlation, and D is estimated if the DZ twin correlation is less than half the MZ correlation. Further details of the twin design, including assumptions, can be found elsewhere An assumption of the twin design is that trait-relevant environments are equally similar for MZ and DZ twins. Tests of this assumption suggest it is valid for personality Descriptives and age effects for the forced-choice mate preferences are shown in Height itself is highly heritable 2 = 56%) and those over 40 (H2 = 39%), but no other preference showed a significant difference between the two groups . The male sample was too small for a similar age comparison.We also tested whether variance components estimates of mate preferences differed between women of normal reproductive age , and those over 40. For hair length preference, twin correlations (and hence variance components) differed significantly (p = .04) between women below the age of 40  varied in the study population, and for all trait preferences this variation was due to significant familial effects in one or both sexes. In general, these familial effects were primarily genetic, with highly significant broad-sense heritability observed for a number of trait preferences. While our design did not afford sufficient power to statistically distinguish between additive and nonadditive genetic effects, it is generally implausible for complex traits to have nonadditive genetic variation in the complete absence of additive genetic variation It is worth noting the strongly sexually dimorphic preferences for height and skin colour see . As to bAcross the different trait preferences, heritability estimates ranged from 6% to 48% in women and from zero to 57% in men. This heterogeneity of variance component estimates was significant in both men and women, but the wide confidence intervals around the individual estimates should be kept in mind when interpreting the findings for particular traits, especially in men. The wide confidence intervals are in part because the measures are dichotomous (and thus imprecise), but are also exacerbated because when only twins raised together are available, since there is limited power to distinguish between family environmental and genetic influences.  Nevertheless, while there were no obvious patterns, it is worth checking for clues by considering which traits preferences had the highest heritabilities. For men, preference for height was easily the most heritable - this appears to only partly reflect the high heritability of height itself, since when individuals’ own height was used as a covariate the heritability of height preference was only a little lower. For women, the two preferences with the highest heritability were for hair length (long/short) and facial hair (beard or moustache/clean shaven), which are presumably the least heritable (within-population) of the target traits . Because these grooming-related traits are environmentally malleable, preferences for them may be under weaker selection, which would be consistent with their high heritability  is expected for genes underlying complex traits Limitations inherent to the classical twin design warrant caution in interpreting our parameter estimates; in particular, separate estimates of additive and nonadditive genetic variance components are imprecise and subject to bias when using only twins, but estimates of the total genetic effect (i.e. broad-sense heritability) should be quite robust


A number of recent works have introduced statistical methods for detecting genetic loci that affect phenotypic variability, which we refer to as variability-controlling quantitative trait loci (vQTL). These are genetic variants whose allelic state predicts how much phenotype values will vary about their expected means. Such loci are of great potential interest in both human and non-human genetic studies, one reason being that a detected vQTL could represent a previously undetected interaction with other genes or environmental factors. The simultaneous publication of these new methods in different journals has in many cases precluded opportunity for comparison. We survey some of these methods, the respective trade-offs they imply, and the connections between them. The methods fall into three main groups: classical non-parametric, fully parametric, and semi-parametric two-stage approximations. Choosing between alternatives involves balancing the need for robustness, flexibility, and speed. For each method, we identify important assumptions and limitations, including those of practical importance, such as their scope for including covariates and random effects. We show in simulations that both parametric methods and their semi-parametric approximations can give elevated false positive rates when they ignore mean-variance relationships intrinsic to the data generation process. We conclude that choice of method depends on the trait distribution, the need to include non-genetic covariates, and the population size and structure, coupled with a critical evaluation of how these fit with the assumptions of the statistical model. Traditional approaches to mapping genes affecting quantitative traits have focused on identifying loci for which an allelic substitution shifts the phenotype of interest in a particular direction . Such quantitative trait loci (QTL) can be described as mean-controlling, because they are primarily observed to affect the expected, ie, average, value of individuals’ phenotypes. It is also possible, however, to detect genetic loci whose allelic substitutions are associated with an increase in variability of the phenotype about its expected value.Figure The biology that explains a vQTL signal can be profound, although its nature will depend on context and may require further modeling and/or experiments to characterize even broadly. Simplistically: a vQTL alerts the researcher to the presence of unmodeled statistical interactions associated with the locus. These could include interactions of potentially high order with other genes or environmental factors, and may implicate a pivotal role for the locus in maintaining phenotypic robustness and/or variability in the face of changing environment, background genetics, and temporal progression. Yet as deep as the implications of such vQTL signals may be, a crucial practical concern is how to detect them in a manner that is powerful, reliable, and robust not only to assumptions about statistical distributions but also to known features of an experiment or population that could potentially bias inference. Herein, we separate two issues: interpretation of vQTL, and statistical detection of vQTL. Both have received a recent surge of interest in the genetics literature -10. We dA number of recent studies -10 have Drosophila melanogaster. Paré et al. [Arabidopsis thaliana located on the exon of MOT1. Although the level of molybdenum in Arabidopsis is known to be regulated by the mitochondrial molybdenum transporter encoded by the MOT1 gene [Rattus norvegicus with normocalciuric Wistar-Kyoto rats, with evidence of sex-specificity for the presence of such effects.Although little is known about how common variance-controlling genes are in the genome or the magnitude of their contribution to trait variation in natural populations , clear eé et al.  found vQé et al. , Shen eté et al.  detectedOT1 gene ,15, no sOT1 gene . Perry eOT1 gene  found seOne of the most exciting developments in this area is the use of methods that identify vQTL as a means to detect interactions. Hereby, epistasis can be detected using a fast search algorithm in one-dimensional parameter space, and G×E can be detected without the need to measure the interacting environmental effect ,4. We 11 have alA current challenge is that statistical methods for detecting vQTL are immature relative to those for detecting mean-controlling QTL. This is in part because estimating effects on the variance is more difficult than on the mean , requiriin silico mapping of vQTL for gene expression in 19 strains of mice. As non-parametric methods, both tests makes few assumptions about the sampling distribution of the data and so are robust to model misspecification [Conover et al.  provide fication ,19.Nonetheless, both tests  require that the data can be grouped into genotype classes, and they lack a natural basis for inclusion of continuous covariates. It is often necessary in GWAS and QTL studies to control statistically for non-genetic covariates such as age, body weight, and other continuous or ordered measurements, including effects that control for uneven relatedness of individuals. Indeed this can be essential to, for instance, increase power or model confounder bias. Such concerns are no less relevant in vQTL detection. The fact that methods based on Levene’s test ,4  that ing et al. . DGLM esr Model; ), and thVisscher & Posthuma  derived http://www.genabel.org), which fits a linear model to the squared residuals from an ordinary GWAS. These approximations trade parametric justifications for convenience and speed: when perfectly estimated, the squared residuals from the initial fit will be gamma distributed (as in the model by Visscher & Posthuma) and therefore violate the normality assumptions required by the second fit. Moreover, like the model of Visscher & Posthuma, these two methods do not account for the fact that the residuals are themselves estimated with error.A less parametrically justifiable but more computationally convenient approach for incorporating an arbitrary set of predictors into a test for effects both in the mean and variance is to fit first a linear model for mean effects, and then fit a separate second linear model on some function of the residuals. Recently, Perry et al.  did thisn = 10,000 observations are drawn from a phenotype model affected by both a SNP and an interacting factor, which in combination manifest as a vQTL. Specifically, we simulated the phenotype of individual i=1,…,n as yi∼N, drawing genotypes as gi∼Bin, and drawing the interacting factor as Fi∼N, with main effect βF=0.85 and interaction βgF=0.06 (using their notation). In this setting, where the sample is large and the genotypes are relatively well balanced, SVLM and DGLM produced almost identical p-values (correlation 0.9996) and the power at a 95% significance level was 0.82 for both methods, whereas Levene’s test produced substantially different p-values and reduced power (0.69). The false positive rates (FPR) were also assessed for the different methods for simulated standard normal trait values, yi∼N, showing very small or no inflated FPR  two-step approach, such as SVLM, have the disadvantage that they involve conditioning on an unknown . EffectsHis the matrix of linear transformation between the observed and fitted response vector  such that hii, where hii are the diagonal elements of H, referred to as “hat values” , and is typical in GWAS where genotypes are imputed. In ordinary QTL analysis and GWAS, genotype probabilities can be used in place of the observed genotypes (possibly after reformulation) as predictors in a linear model of the phenotype -26. In sA common problem in GWAS is accounting for effects of population structure. We  have proOne of the greatest challenges in the vQTL detection is how to choose the scale of measurement on which to draw conclusions about estimated effects. When we detect a SNP whose alleles increase both the mean and the variance, should we interpret it as a vQTL with a significant marginal effect, or a mean-controlling QTL for a trait that was analyzed on the wrong scale? Suppose, for instance, that we had a cylindrical organism (such as a snake or a worm) whose body width increases with its body length, and we have a gene with a strong additive effect on body length, which is itself normally distributed. A GWAS on the volume of this cylindrical organism is likely to detect a QTL controlling both mean and the variance, despite the fact that a simpler explanation is available.To circumvent such ambiguities, some researchers have used the coefficient of variation  to detect genotypic effects on the variance . This apIn some cases, the trait is known to be well approximated by a distribution that has a known mean-variance relationship, such as the Poisson or gamma distribution. In such cases, it will often be preferable to model that distribution explicitly and define vQTL parameters as those that alter the variance in a way not already anticipated by concomitant effects on the mean. DGLMs provide a natural basis for such models when the known sampling distribution is member of the exponential family .Yi of individual i be distributed as Yi∼Poisson(μi), where the individual’s genotype gi affects the phenotype through the relation μ=−1,βg=0.05. This set up is similar to that used in Struchalin et al. [To illustrate the effect of a misspecified distribution on vQTL detection, we consider the following simulation study in which vQTL-detection methods are applied to a non-normal phenotype. Let the phenotype n et al.  with theOne interesting possible cause of a vQTL signal arises when a QTL affecting a primary trait of interest also affects another (secondary) trait in a way that can restrict variation of the primary trait . Cheverud et al.  develop We further note that several semi-parametric and fully parameterized estimation methods have been developed for animal breeding purposes over the last couple of decades. These methods incorporate the additive relationships between individuals to assess the possibility of reducing phenotypic variability in breeding programs. They are not directly applicable in GWAS nor QTL analysis but the modeling approach is very similar to that of vQTL detection. A future possibility is to perform standard GWAS using estimated breeding values for variability as response. A deeper understanding of new methods for vQTL detection could therefore be obtained by relating these to the already existing literature on genetic heterogenity in animal breeding  or random polygenic effects. Parametric methods fully account for the uncertainty of fitted parameters in both the mean and the variance parts of the model, and also allow fitting of covariates and random polygenic effects in both parts, but are more sensitive to modeling assumptions. Semi-parametric or two-stage approaches can be faster but come at the price of shortcuts that in some cases can lead to bias. The choice of method depends on the trait distribution, the need to include non-genetic covariates, and the population size and structure. We advise that the assumptions of the chosen model be evaluated and compared with those of alternatives, and we expect that if this is performed in a careful manner then these methods could be of great use in both the analysis of GWAS and QTL mapping data.Both authors wrote the text and approved the final version of the manuscript.


Nature Genetics. 2005, 683–691) proposed an interesting two-stage test for genome-wide association (GWA) studies under family-based designs which performs genomic screening and replication using the same data set. In the first stage, a screening test based on the between-family information is used to select markers. In the second stage, an association test based on the within-family information is used to test association at the selected markers. However, we learn from the results of case-control studies  that this two-stage approach may be not optimal. In this article, we propose a novel two-stage joint analysis for GWA studies under family-based designs. For this joint analysis, we first propose a new screening test that is based on the between-family information and is robust to population stratification. This new screening test is used in the first stage to select markers. Then, a joint test that combines the between-family information and within-family information is used in the second stage to test association at the selected markers. By extensive simulation studies, we demonstrate that the joint analysis always results in increased power to detect genetic association and is robust to population stratification.In family-based data, association information can be partitioned into the between-family information and the within-family information. Based on this observation, Steen et al. ( R “top” markers. In the second stage of FTSA, a family-based association test based on within-family information is used to test the R selected markers for association. FTSA is robust to population stratification because the association is determined by the family-based association test in the second stage. Furthermore, since the statistic used in the first stage is statistically independent of that in the second stage, the overall significance level of the algorithm does not need to be adjusted for the first stage. In the following discussion, we call the tests used in the first stage and in the second stage screening test and association test, respectively.Currently, the family-based association tests such as the TDT and its extensions In case-control studies, several authors have proposed a two-stage design which utilizes two independent samples λ can be estimated using genotypes at genomic markers In case-control studies, it has been long recognized that population stratification can seriously confound association results In this article, we propose a novel approach to do joint analysis within the framework of FTSA. We first perform PC analysis based on parental genotypes at a set of genomic markers and then use a PC approach to eliminate any effect of population stratification both in genotypes at the candidate marker and trait values for all family members. A screening test is then constructed based on adjusted between-family information and parental trait values. We use this screening test which is robust to population stratification to select markers in the first stage. In the second stage, we do joint analysis i.e. use a test that is a combination of the screening test and the association test to test association at the selected markers. The joint analysis is robust to population stratification because both the screening test used at the first stage and the joint test used at the second stage are robust to population stratification. We evaluate the performance of our joint analysis approach by simulation studies under a variety of population admixture models. Our simulation studies show that the proposed joint analysis approach is robust to population stratification and is consistently more powerful than FTSA proposed by Steen et al. L markers have been genotyped for each sampled individual. For the K = 1 and 2 for the two parents), where genotypic score is the number of copies of minor allele.Consider a GWA study of l markers from the L markers in GWA panel as genomic markers. For the l randomly chosen markers of the k = 1 and 2 for the two parents). We perform a PC analysis to summarize the genotype data at genomic markers. Because our data are family data, a naive PC analysis with all available data will result in biased directions of maximum variability for the data. Thus, the PC analysis is applied to only the parents in each family.We assume that the parental phenotypes are available. In this case, by incorporating parental phenotypes, Feng et al. K PCs . Because the PCs represent the genetic background information, we adjust both the trait and genotype at candidate loci for this genetic background information by applying linear regression k = 1,2. Let Let Based on the adjusted trait values and genotypic scores, we propose the following screening test, called admixture screening (Ascreen) test,We use quantitative pedigree transmission disequilibrium (QPTD) as our family-based association test C such that a marker is selected if T denote the event of In the first stage, we select R markers with the smallest p-values of the admixture screening test. This selection means that there is a constant We compare the proposed joint analysis method with two other methods that is described below. One is FTSA proposed by Steen et al We used simulation studies to compare the performance of the joint analysis with FTSA and AFTSA. We also compared FTST with AFTSA to see if adjusting population stratification in the screening test can improve power of FTSA. The simulation setup used in this study was similar to that of Zhu et al In this set of simulations, we simulated samples based on the haplotype data of 120 European chromosomes (CEU) released by the HapMap project To generate trait values under the null hypothesis, for a nuclear family with m children, let ρ . The conditional distribution of This covariance structure meant that the father and mother were independent, and parents with children and children with children were correlated with the correlation coefficient of B members from model g at the disease locus andUnder the alternative hypothesis, we generated the trait values of a nuclear family with a and A were the two alleles at the disease locus and A was the high risk allele); β was a constant and β was determined by heritability h and was given by p was the allele frequency of the high risk allele.for a dominant, recessive, and additive model, respectively  and 120 African chromosomes (YRI) released by the HapMap project μ was a constant that measured the difference of the average trait values between the two subpopulations; h and the relationship was given in simulation set 1. We used the same value of h in the two subpopulations and thus h = 0 and thus h>0 for generating the trait values under the alternative hypothesis.if this family was from European subpopulation andAgain, we simulated samples based on the chromosome 1 data of 120 European chromosomes and 120 African chromosomes released by the HapMap project μ and β were constants and β can be determined by heritability h. Again, we set h = 0 for generating the trait values under the null hypothesis, and h>0 for generating the trait values under the alternative hypothesis.We generated the trait values of a nuclear family with μ. However, we fixed the value of R, the number of markers selected at the first stage, as 10. We evaluated type I error rates of the three methods  as well as two screening tests In all of the three sets of simulations, we used 1000 replicated samples to evaluate the type I error rates and power and considered nuclear family with one child i.e. trio as the family structure. To evaluate the type I error, we considered different sample sizes, different number of markers used to control for population stratification, and different values of pulation , it woulpulation , 3. Tablpulation  and it wpulation , 3. The h, different number of markers selected at the first stage, different values of μ, and different sample sizes. To evaluate power, in each replication, we randomly chosen a marker with minor allele frequency  in the interval  as the disease locus and the minor allele as the high risk allele for dominant and additive models while the major allele as the high risk allele for recessive model. Results of power comparison were summarized in For power comparison, we considered different scenarios which included different values of heritability l 0.1, 0. as the dIn this article, we proposed a novel method to perform joint analysis within the framework of the family-based two-stage analysis. In the joint analysis, we first constructed a screening test that was based on between-family information and was robust to population stratification. In the first stage, we used this screening test to select markers. In the second stage, we did joint analysis i.e. used a test that was a combination of the screening test and the association test to test association at the selected markers. The joint analysis was robust to population stratification because both the screening test and the association test are robust to population stratification. Our simulation studies showed that the joint analysis was consistently more powerful than two-stage approaches in which the association test used in the second stage was only based on within-family information.Although we have discussed the joint analysis, in which we only tested the selected markers in the second stage, it is straightforward to extend the joint analysis to the p-value weighting scheme It should be noted that the PC approach used in R, the number of markers selected in the first stage. Although there is no unique answer in choosing an optimal value of R, our simulations indicate that 10 is a good choice of R which is consistent with the results of Steen et al R in general.One remaining question is choosing the value of


Next-generation sequencing (NGS) technology has provided researchers with opportunities to study the genome in unprecedented detail. In particular, NGS is applied to disease association studies. Unlike genotyping chips, NGS is not limited to a fixed set of SNPs. Prices for NGS are now comparable to the SNP chip, although for large studies the cost can be substantial. Pooling techniques are often used to reduce the overall cost of large-scale studies. In this study, we designed a rigorous simulation model to test the practicability of estimating allele frequency from pooled sequencing data. We took crucial factors into consideration, including pool size, overall depth, average depth per sample, pooling variation, and sampling variation. We used real data to demonstrate and measure reference allele preference in DNAseq data and implemented this bias in our simulation model. We found that pooled sequencing data can introduce high levels of relative error rate  and that the error rate is more severe for low minor allele frequency SNPs than for high minor allele frequency SNPs. In order to overcome the error introduced by pooling, we recommend a large pool size and high average depth per sample. Over the last decade, large-scale genome-wide association studies (GWAS) based on genotyping arrays have helped researchers to identify hundreds of loci harboring common variants that are associated with complex traits. However, multiple disadvantages have limited genotyping arrays' ability for disease association detection. A major disadvantage of genotyping arrays is the limited power for detecting rare disease variance. Rare variants with minor allele frequency (MAF) less than 1% are not sufficiently captured by GWAS . The mosMost of the above limitations can be overcome by using high throughput NGS technology . NGS canThe concept of pooling in genetic studies began in 1985 with the first genetic study to apply a pooling strategy . Since tUsually two different kinds of pooling paradigms are involved. The first is multiplexing . On an Illumina HiSeq 2000 sequencer, one lane can generate, on average, from 100 to 150 million reads per run. For exome sequencing, from 30 to 40 million reads per sample are needed to generate reliable coverage in the exome for variant detection. Thus, the common practice is to multiplex from 3 to 4 samples per lane to reduce cost. Using multiplexing with barcode technology, we are able to identify each read's origination. The disadvantage of multiplexing with barcoding is the extra cost of barcoding and labor. The cheaper alternative to pooling with multiplexing is pooling without multiplexing, which prevents us from identifying the origin of each read.In this study, we focused on pooling without multiplexing. By using comprehensive and thorough simulations, we tried to determine the effectiveness of estimating allele frequency from pooled sequencing data. In our simulation model we considered important factors of pooled sequencing, including overall depth, the average depth per sample, pooling variation, sampling variation, and targeted minor allele frequency (MAF). Another important issue we addressed in our simulation is the reference allele preferential bias, which is a phenomenon during alignment when there is preference toward the reference allele. We used real data to show the effect of reference allele bias and adjusted our simulation model accordingly. We describe our simulation model in detail and present the results from the simulation.F be the true allele frequency in the pool. Under the ideal assumption, all samples' contributions to the pool are equal. However, in practice, factors such as human error and library preparation variation can affect a sample's contribution to the pool. Very likely, each time a sample is added to the pool, an error is introduced. We let εi denote the error of sample i during the pooling process, and εi should follow a normal distribution N, where µ = 0, and σ2 denotes the variance of error in the pool. We assume that the amount of DNA added to the pool for each sample c + εi follows a normal distribution N, where c is a constant and denotes the ideal constant contribution to the pool by each sample. The probability that a read is contributed by sample i can be represented as pi = (c + εi)/∑i=1Ns(c + εi), where Ns denotes the number of samples in the pool. The contribution of each sample in the pool to a SNP can be modeled as a multinomial distribution R1, R2, R3, …, RNs ~ Multinomial, where D equals the depth at this SNP and Ri represents the reads contributed by sample i for this SNP. The depth D follows a possion distribution Poisson(λD), where λD equals the average depth for the exome regions. For sequencing data, the reads at heterozygous SNPs should have an allele balance of 50%, meaning 50% of the read should support the reference allele while the other 50% of the read should support the alternative allele. Thus the reads that support the alternative allele should follow a binomial distribution Binomial.We designed a thorough simulation model to closely reflect the real-world pooled sequencing situation. Our simulation model includes notations which we have defined as follows: let In our study we estimated the average depth for the exome regions as follows:In general the read output for 1 lane on an Illumina HiSeq 2000 sequencer is around 120 million reads. The most popular exome capture kits including Illumina TruSeq, Agilent SureSelect, and NimbleGen SeqCap EZ capture almost 100 percent of all known exons (about 30 million base pairs). Most capture kits claim that they have capture efficiency of at least 70 percent, but, in practice, it has been shown that the capture efficiency of all these capture kits are only around 50 percent , which in is the number of simulations we performed to estimate the target allele frequency F. In our simulations, we set n = 10,000. Unlike the traditional RMSE, we divided it by the target allele frequency F to make the result relative to the allele frequency we were simulating, so we could compare RMSE for allele frequencies as small as 0.5% and as large as 50%.To measure the accuracy of the allele frequency Binomial to Binomial where P follows a normal distribution N, where µP and σp2 are estimated by the empirical mean allele balance we observed in real data.Reference allele preferential bias is a phenomenon during alignment when there is preference toward the reference allele. Degner et al. described such bias in RNA-seq data . To examThree simulations were conducted to evaluate the accuracy of allele frequency estimation from pooled sequencing data. The detailed descriptions of the three simulations are as follows.ε and relative RMSE under different pool sizes  and different MAF . Each sample's DNA contribution c + εi to the pool follows a normal distribution N. For simulation purpose, we set an arbitrary value c = 10 units; the actual value of c does not affect the outcome of the simulations, because the simulation merely scales around it. To best represent the scenario in practice, we used several different standard deviations values for the distribution of sample contribution to the pool. For the ideal situation, we set σ2 to a very small number (10−5); then, we increase σ2 to 1, 2, and 4  to see the effect of larger error variance on the accuracy of allele frequency estimation using pooled sequencing data. Each allele frequency was simulated 10,000 times.The goal of λps = λD/Ns . If the average depth of exome regions for the pool of 200 people is 600x, then the depth per sample is 3x. In this simulation, we used λps = 0.5, 1, 2, 4, 6, 8, 10, 12, 14,16, 18, and  20, pool size Ns = 200, 400, 800, and 1600, and MAF = 0.5%, 1%, %5, 10%, 20%, 30%, 40%, and 50%. Each allele frequency was simulated 10,000 times.The goal of The goal of N. In an ideal situation, σ2 is small, and if we fix overall depth, the pool size does not make a significant difference for the RMSE. For example, in an ideal situation, for MAF = 0.5, the relative RMSE for pool size Ns = 200, 400, 800, and 1600 equals 0.023, 0.024, 0.024, and 0.020, respectively. However, if we increase σ2, pools with greater size tend to have lower RMSEs. For example, when σ2 is increased to 4, for MAF = 50%, the relative RMSEs for pool size Ns = 200, 400, 800, and 1600 equals 0.028, 0.0250, 0.023, and 0.022, respectively. Increasing σ2 clearly also increased relative RMSE for all MAFs and for all pool sizes. For example, for pool size Ns = 200, MAF = 50%, σ2 = 0.00001, 1, 2, and 4, the relative RMSE are 0.020, 0.021, 0.023, and 0.028, respectively. Also lower MAF tended to have high relative RMSE than high MAF. For example, in an ideal situation, for pool size Ns = 200, the relative RMSEs for MAF = 0.5%, 1%, 5%, 10%, 20%, 30%, 40%, and 50% are equal to 0.289, 0.202, 0.088, 0.061, 0.041, 0.031, 0.030, and 0.020, respectively. The results of We assume that each sample's DNA contribution to the pool follows a normal distribution λps and pool size Ns. We found that, with the same average depth per sample λps, higher pool sizes will generate lower relative RMSEs. Also, as the MAF increases, the relative RMSE decreases. For example, for MAF = 50% and average depth per sample λps = 1, the relative RMSEs for Ns = 200, 400, 800, and 1600 are 0.071, 0.050, 0.036, and 0.025, respectively. If we can infinitely increase pool size or average depth per sample while fixing the other, the RMSE will reach zero. The result of In this simulation, the goal was to examine the relationship between average depth per sample In our study, we performed simulations at each MAF 10,000 times. However, in practice, we do not have the resources to measure a SNP 10,000 times and then take the average. In real exome sequencing, each SNP is only measured one time. Ns = 1092 1000 times and computed the median error rate for each simulation (In this simulation, we simulated the scenario of pooled exome sequencing. Using data from the 1000 Genomes Project as prior information that contains genotyping data from 1092 individuals, we built an empirical distribution of MAF . Based omulation . The resFinancial Implication. The ultimate goal of pooling is to ease the financial burden on large association studies. Based on the most up-to-date pricing information on NGS, we compared the total cost of conducting association studies using pooling at different pool sizes with individual sequencing using Illumina HighSeq 2000 sequencer, which contains 2 flow cells, and each flow cell contains 16 lanes. Our simulation showed that there are several important factors to consider when designing a pooling study. Those factors include sample size, targeted MAF, and, most importantly, the depth. The sample size directly affects the ability to detect rare SNPs. Larger pool size will increase the accuracy of MAF estimation with the same per sample depth but will not have much effect with the same overall depth. Similarly, with the same pool size, increasing depth will decrease relative RMSE. Our simulation also showed that pooled sequencing is not ideal for estimating the MAF of rare SNPs. The relative RMSE is much higher for SNPs with MAF < 1% compared to SNPs with MAF > 5% .Sequencing pooled DNA will ease financial burdens and make large association possible. At the same time, however, pooling introduces additional errors. A majority of the errors are caused by the unequal representation of each sample's DNA in the pool. This unequal representation could be due to human or machine error, which we have considered in our simulation. There are other factors which can also cause the unequal representation, such as a sample's DNA quality and variation introduced in the PCR/amplification stage. Unfortunately, we can only minimize such errors and variation using more sophisticated lab techniques. Even if every sample is equally represented in the pool, the sequenced data still do not truly reflect the equality due to sampling variance. Based on our simulation results, when designing a pooling study, we recommend the following: larger pool size is better, and higher depth is better. More elaborately, it is better to keep balance between pool size and depth. We recommend keeping the average depth per sample at 10 minimum if rare SNPs are not of interest; otherwise, average depth per sample at 20 minimum is highly recommended.


X). However, SNP calling and allele frequency estimation in such studies is associated with substantial statistical uncertainty because of varying coverage and high error rates.Estimation of allele frequency is of fundamental importance in population genetic analyses and in association mapping. In most studies using next-generation sequencing, a cost effective approach is to use medium or low-coverage data  accuracy of allele frequency estimation, (2) accuracy of the estimation of the distribution of allele frequencies across neutrally evolving sites, and (3) statistical power in association mapping studies. Using real re-sequencing data from 200 individuals obtained from an exon-capture experiment, we show that the patterns observed in the simulations are also found in real data.Overall, our results suggest that association mapping and estimation of allele frequencies should not be based on genotype calling in low to medium coverage data. Furthermore, if genotype calling methods are used, it is usually better not to filter genotypes based on the call confidence score. D [D [H [ST F[The frequency of an allele in the population is a fundamental quantity in human statistical genetics. This quantity forms the basis of many population and medical genetic studies. Many evolutionary forces change allele frequencies. Consequently, allele frequencies can be used to infer past evolutionary events. For example, allele frequencies at single nucleotide polymorphisms (SNPs) can be used to infer the demographic history of a population ,2. PatteD , Fu and D [D , Fay andD [D [H  and FST [H [FST  are dire[H [FST -14. In p[H [FST -17.Given the importance of allele frequencies in genetic studies, it is critically important to be able to estimate them reliably. Traditionally, allele frequencies were simply estimated by counting the number of times each allele had been seen in a sample from the population. This approach was often successfully used on SNP genotype data and Sanger sequencing data because the genotypes for each individual could often be unambiguously determined. However, this approach may fail when applied to data from next-generation sequencing technology. First, next-generation sequencing data has a higher error rate than traditional Sanger sequencing or SNP genotyping assays -20. SecoSeveral different approaches have been proposed to attempt to make accurate inferences of allele frequency from next-generation sequencing technologies -31. It iHere we discuss the properties of a new likelihood approach designed to estimate the population minor allele frequency from next-generation sequencing data. We show that the new likelihood method can obtain accurate estimates of allele frequencies, even when the depth of coverage is quite shallow. Further, we show that the new likelihood method either performs as well as, or better than, genotype calling methods. Finally, we discuss the performance of the likelihood approach in testing for differences in allele frequency between cases and controls.The minor allele is the less frequent allele in the population at a variable site. We first describe two main approaches to estimate the minor allele frequency (MAF) at a particular site in the genome. The first approach involves inferring individual genotypes and treating those inferred genotypes as being completely accurate when estimating the MAF. We then examine the performance of a likelihood framework that directly takes the uncertainty in assigning genotypes into account. Throughout our work, we assume that all segregating sites are biallelic.One way to estimate the MAF from next-generation sequencing data is to first call a genotype for each individual using sequencing data, and then use those genotypes as if they are the true ones. This was the approach traditionally used for genotype data and Sanger sequencing data. It is not clear how well it will perform when applied to next-generation sequencing data.j, for each individual i, the likelihood for each of the three possible genotypes  is given as:A maximum likelihood approach can be used to infer the genotype for each individual from the next-generation sequencing data. At each site i,j Dis the observed sequencing data in individual i at site j, ig,j ∈ {0, 1, 2} is the number of minor alleles contained in the genotype of each individual, and j taking the read quality scores into account. This is represented as the genotype likelihood and is found in the genotype likelihood file (GLF) which is produced in many programs that analyze next-generation sequencing data, such as SOAPsnp and MAQ [where  and MAQ ,32.gk)  if f is 1, indicating that the most likely genotype is at least 10 times more likely than the second most likely one. Note that this type of filtering may result in higher confidence for the "called" genotype, but it also results in more missing data.To assign a genotype to a particular individual, the likelihood of each of the three possible genotypes can be calculated for the individual. The genotype with the highest likelihood can then be assigned. However, researchers often prefer a more stringent calling criterion and will not assign a genotype to an individual unless the most likely genotype is substantially more likely than the second most likely one. Here the three possible genotypes are sorted by their likelihoods: i is obtained by summing over the probabilities corresponding to all three possible genotypes.Instead of estimating the MAF from the called genotypes, a maximum likelihood (ML) method introduced by Kim et al.  . Then, assuming independence among individuals, the likelihood of the MAF at this locus is a product of all the likelihoods computed across all N individuals:Suppose that the three genotype likelihoods defined in Equation 1 are available. Using the same notation as above, let j pcan be computed either by directly maximizing the likelihood for a restricted parameter space using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method [The ML estimate of ) method -40 or by) method ,41. WhenIn practice, often the second most common nucleotide across individuals can be used as the minor allele. However, for rare SNPs , it is hard to determine which allele is the minor allele, since all four nucleotides may appear in some reads due to sequencing errors. To deal with this situation, we now describe a likelihood framework that takes the uncertainty in the determination of the minor allele into account.j we know the major allele M. Note that deciding which of two common alleles is likely to be the major one is not important since we are mostly concerned with estimating the frequencies at rare SNPs. Further, for alleles with intermediate frequencies (around 50%), the distinction between major and minor allele is less important. Assign the other three non-major nucleotides m1, m2, and m3. The likelihood introduced in Equation 2 assumes a fixed major allele M and fixed minor allele m. Therefore, to allow for uncertainty in the designation of the minor allele, the likelihood function can be modified as:Suppose that for site Further, assuming that any of the three possible minor alleles is equally likely, we obtain:l(1), l(2), l(3)), where l(1) is the largest one. Then,where G-test:In association studies, SNPs showing significant differences in allele frequency between cases and controls are said to be associated with the phenotype of interest. Association mapping can be performed using data from next-generation sequencing studies. We first discuss approaches that require calling individual genotypes and then perform a test for association using the called genotypes. In this approach, a genotype is first called for each individual. The genotypes can be filtered or unfiltered. Assuming independence across individuals and HWE, a 2 × 2 contingency table can be built by counting the number of major and minor alleles in both the cases and controls. This leads to the well-known likelihood ratio test for independence, the k,h Ois the frequency observed in a cell, and k,h Eis the frequency expected under the null hypothesis in which the allele frequency is the same between cases and controls. The well-known Pearson's chi-square test is asymptotically equivalent to the G-test. If the table is generated from true genotypes, then the G-statistic asymptotically follows a chi-square distribution with 1 degree of freedom (χ2(1)). However, in our studies, we construct the G-statistic using "called" genotypes, thus HWE may not hold due to over- and under-calling of heterozygotes. Furthermore, constructing the test statistic by counting "called" genotypes instead of "observed" genotypes likely introduces extra variability. Therefore, the statistical theory may not be valid any more. Note that when a genotype is not called for a certain individual, the data is considered missing and is not included in the 2 × 2 table.where j whether the allele frequency is the same between cases and controls. Formally, we compute the likelihood of the hypotheses O H: jp,1 = jp,2 and A H: jp,1 ≠ jp,2, where jp,1 and jp,2 are the MAFs in cases and controls, respectively.Instead of calling genotypes, the likelihood framework allows for uncertainty in the genotypes and tests at each site m) and major (M) alleles are known, the likelihood of the minor allele frequency can be computed as described in Equation 2, and the likelihood ratio test statistic is computed as:Assuming that minor , and the maximum likelihood method (ML). For rare SNPs, the minor allele type is often not apparent. When calling genotypes, the second most common nucleotide is assumed to be the minor allele. The ML method directly incorporates uncertainty in determining the minor allele and unless otherwise stated, results using the unknown minor allele method (Equation 3) are shown. Note that the unknown minor allele ML method performs similarly to the known minor allele ML method but the former better for very rare SNPs , called genotypes without any filtering , called genotypes with filtering f = ; Call F,We first evaluated how well the different approaches were able to estimate the MAF in 200 individuals across a range of sequencing depths for 1,000 SNPs with a true MAF of 5%. Figure The results are dramatically different for the new ML method. This method provides unbiased estimates of the MAF (median of ~4.9%) across a range of depths. Even at 2×, the estimates show only a slightly larger variance than those based on the true genotypes.We also compared the estimated mean squared error . Here we simulated 20,000 SNPs where the distribution of the true MAFs followed the standard stationary distribution for an effective population size of 10,000 (see Methods). Note that in practice, however, it is very difficult to distinguish a very rare SNP from a sequencing error. Therefore, for comparison purpose with real data, we discarded SNPs with estimated MAF less than 2%. Figure As expected, with a high depth of coverage, such as 10× per individual, all methods provide estimated MAF distributions that are similar to the expected distribution based on the true genotypes Figure . With a The picture is entirely different for the ML method. The estimated MAF distribution obtained from the new ML method closely follows the true distribution even with shallow depths of coverage. Here there is almost no excess of low-frequency SNPs. At a depth of 4×, the proportion of SNPs in the second bin of the histogram is 18.4%, which is very close to the expected proportion (18%). Thus, more reliable estimates of the frequency spectrum can be made from low-coverage data by using our likelihood approach than by using the genotype calling approaches.G-test) to our likelihood ratio test (LRT) that accounts for uncertainty in the genotypes. We examine the distribution of the test-statistic under the null hypothesis of no allele frequency difference between cases and controls. We also compare the power of the different approaches.We compare the performance of methods that treat inferred genotypes as true genotypes in tests of association (using a G-statistic and LRT statistic follow a chi-square distribution with one degree of freedom (χ2(1)). Therefore, we have compared the null distribution of the G-statistic computed based on calling methods as well as the LRT statistic to the χ2(1) distribution using QQ-plots  distribution. However, the distribution of the G-statistic computed based on the called genotypes substantially departs from a χ2(1) distribution. Calling genotypes and then treating those genotypes as being accurate produces a vast excess of false-positive signals if the p-values are computed using a χ2(1) distribution. For example, at a depth of 2×, 11% of the SNPs had a p-value less than 5%, compared to the expected 5%. The effect is caused by an increase in the variance, due to overcalling homozygotes as heterozygotes, in the allelic test used here for detecting association. Genotypic tests such as Armitage trend test, which are robust to deviations from Hardy-Weinberg equilibrium, do not show a similar increase in the false positive rate  distribution for either 2× or 5× depths of coverage.With reasonably large sample sizes, standard asymptotic theory suggests that under the null hypothesis both the s Figure . We simuχ2(1) distribution under the null hypothesis, to make a fair comparison, we obtained the critical value for each false positive rate based on the empirical null distribution. The power is computed as the fraction of simulated disease loci that have a statistic exceeding the critical value. Overall, we find that the LRTperforms better than the G-test based on either genotype calling method  curves for each of the different association tests. These curves show the power of the test at different false-positive rates. Since the distributions of some of the test statistics do not follow the d Figure . For exaWe analyzed 200 exomes from controls for a disease association study that have been sequenced using Illumina technology at a per-individual depth of 8× . We usedFirst, we explored the accuracy of the estimates of the MAF from next-generation sequencing data for 50 SNPs by comparing them to the estimated MAFs from Sequenom genotype data. Both the estimates using the ML method and the genotype calling method without filtering are highly correlated with the estimates made from the Sequenom genotype data  in the G-statistic using called genotypes both with and without filtering as well as the LRT statistic. Figure χ2(1) distribution. As seen in simulation studies, the null distribution of the G-statistic calculated when calling genotypes without filtering substantially departs from the χ2(1) distribution. However, the null distribution of the LRT statistic closely follows the χ2(1) distribution. The inflation factor [LRT statistic performs well when applied to real data.Finally, we used this exome-resequencing data to simulate a case-control association study. To examine the distribution of the association test statistics under the null hypothesis, we randomly assigned 100 individuals to a case group and the other 100 to the control group. For all SNPs on chromosome 2 with MAF estimates > 2% , we tested for allele frequency differences between cases and controls by computing the n factor  is 1.01,The likelihood method discussed here is an extension of our previous approach  which waThough not surprising, it is important to note that with higher sequencing coverage, the particular approach used to estimate allele frequencies does not matter as much. For depths of coverage > 10×, the genotype calling methods both with and without filtering behave appropriately and similarly to the ML approach. Thus, with high depths of coverage, the traditional and simple method of calling genotypes and then treating those genotypes as being known with certainty is still effective. The reason for this is that with such high depth, the called genotypes are likely to be accurate. With lower depths of coverage, however, there is considerable uncertainty regarding the true genotype. Often the most-likely genotype will not be the true genotype, leading to biases in estimates of allele frequency and spurious signals of association in case-control studies. In this situation, the ML method is a superior approach.In our simulations, we compared the performance of our ML approach to a relatively simple genotype calling approach (see Methods). It is possible that more sophisticated genotype calling approaches such as SOAPsnp , MAQ 2323, and GWe have explored whether it is better to call genotypes with filtering or without filtering when analyzing low-coverage data. Intuitively, one would expect that if there was uncertainty in the genotypes, it would be better to call genotypes only if one was very confident in that genotype and treat the other less confident genotypes as missing data. However, as discussed by Johnson et al. , callingStudies have suggested that genotype calling approaches that use LD information to call genotypes ,36 may rH0 : p = 0) and under the hypothesis that there is a SNP at a given site (A H: p > 0). Such an approach is the subject of ongoing research. A feature of our approach to estimating frequencies is that it is not necessary to first call SNPs prior to estimating frequencies, which may be beneficial in certain circumstances.As currently implemented our method does not tackle the problem of SNP calling itself. In principle, our approach could be extended to use a LRT to call SNPs. Specifically, the test could compare the probability of the data under the hypothesis that there is no SNP at a given site  since it models the uncertainty in assigning individual genotypes. However, genotype calling approaches can lead to biased inferences when applied to low-coverage data. We have also extended the likelihood approach to test for differences in estimated minor allele frequency between cases and controls. Through simulations and the analysis of exomes from 200 individuals, our LRT has appropriate false-positive rates and higher power than genotype calling approaches when analyzing low-coverage data. Finally, we have shown that under certain circumstances, if one uses genotype calling approaches, it is better to not filter genotypes based on the call confidence score.We performed extensive simulation studies to compare the performance of the likelihood methods with methods based on called genotypes. Specifically, we simulated data to assess (1) the accuracy of the estimates of the MAF, (2) the accuracy in estimating the distribution of MAFs across genome, and (3) statistical power in association mapping studies. Due to computational constraints, we simulated SNPs in the sequencing data directly rather than simulating raw sequencing reads.We simulated SNPs with a specified MAF, number of individuals and per-individual sequencing depth. When simulating causal SNPs in association studies, MAFs for cases and controls were assigned using a multiplicative disease model. For this model, the prevalence of the disease was fixed at 10%. We examined two sets of MAFs and relative risks. First, the combined MAF in cases and controls was 1% and the relative risk was 2. Second, the combined MAF was 5% and the relative risk was 1.5. As an example, with a combined MAF of 1% and a relative risk of 2.0, the obtained MAFs for cases and controls are 1.98% and 0.89%, respectively. Each individual genotype was simulated assuming Hardy-Weinberg equilibrium with the given MAF. Read bases were then generated for each individual by copying each allele a Poisson-distributed number of times with mean equal to the half the specified individual depth. Each read base then may have been altered to one of the other three nucleotides at a specified type-specific sequencing error rate. The type-specific error rates used to simulate the data were estimated (see below) from 200 exomes sequenced using the Illumina platform . The simWe also evaluated the performance of the approaches to estimate a distribution of MAFs when the specified allele frequencies were drawn from a the stationary distribution under a Wright-Fisher model with a population size of 10,000. Under such a model, population allele frequencies are proportional to 1/x, where x is the frequency of the allele in the population . Given tVarious methods have been proposed to compute genotype likelihoods from next-generation sequencing data, which recalibrate quality scores of read bases and attempt to correct for sequencing error structures and other complexities in the genome ,32. HoweinA,j, inC,j, inG,j, inT,j) in each individual i at site j. Denote the twelve base-specific sequencing error rates as bε,b' = P, where b, b' ∈ {A, C, G, T} and b ≠ b. Given these error rates, the likelihood for genotype g is computed as follows:The data consist of the observed counts of each of the four nucleotides . Note that the counts of four nucleotides are re-ordered as counts of minor, major, and other two nucleotides , and the set of error rates is also re-arranged to h, h' ∈ {m, M, o1, o2} for each site. These genotype likelihoods were computed from our simulated datasets. The globally-determined base-specific sequencing error rates were set to the true values used to simulate the data , more than 170 individuals satisfy this condition. Second, we examined the proportion of SNPs with different frequencies when using different strategies to estimate the MAFs. Finally, we used these data to simulate a case-control association study by randomly assigning 100 exomes to a case group. We then examined the behavior of the different test statistics under the null hypothesis.All the source code used for our simulation studies, estimation of parameters, and tests of association are publicly available . Then each boxplot was drawn using estimated MAFs with known (left) and unknown(right) minor alleles.Click here for fileχ2(1) distributionQQ-plot comparing the null distribution of the Armitage trend test statistic with a . QQ-plots comparing the null distribution of the test statistic of interest with a χ2(1) distribution. The first three columns correspond to the Armitage trend test statistic computed using the true genotypes (True), called genotypes without filtering , and called genotypes with filtering , respectively. The fourth column corresponds to the likelihood ratio test statistic with unknown minor allele (LRT). Assuming 500 cases and 500 controls, under the null hypothesis, a set of 5,000 sites were simulated with a MAF of 5% with a sequencing depth of 2× (upper panels) and 5× (lower panels). The "Inflation" factor [" factor  is shownClick here for fileReceiver operating characteristic curve of the Armitage trend test. Receiver operating characteristic (ROC) curves of four tests of association. For the definition of the four statistics, see the caption of Additional file Click here for fileEstimates of type-specific sequencing error rates. Type-specific sequencing error rates estimated from 200 exomes [0 exomes  using ouClick here for fileManual of our programs: simreseq and testassoc. Manual of our programs: simreseq and testassoc.Click here for fileSource code of our programs: simreseq and testassoc. All the source code of our programs used for the simulation studies, estimation of parameters, and tests of association.Click here for file


Most of previous empirical studies with genome-wide prediction were focused on within-environment prediction based on a single-environment (SE) model. In this study, we evaluated accuracy improvements of across-environment prediction by using genetic and residual covariance across correlated environments. Predictions with a multienvironment (ME) model were evaluated for two corn polygenic leaf structure traits, leaf length and leaf width, based on within-population (WP) and across-population (AP) experiments using a large maize nested association mapping data set consisting of 25 populations of recombinant inbred-lines. To make our study more applicable to plant breeding, two cross-validation schemes were used by evaluating accuracies of (CV1) predicting unobserved phenotypes of untested lines and (CV2) predicting unobserved phenotypes of lines that have been evaluated in some environments but not others. We concluded that (1) genome-wide prediction provided greater prediction accuracies than traditional quantitative trait loci-based prediction in both WP and AP and provided more advantages over quantitative trait loci -based prediction for WP than for AP. (2) Prediction accuracy with ME was significantly greater than that attained by SE in CV1 and CV2, and gains with ME over SE were greater in CV2 than in CV1. These gains were also greater in WP than in AP in both CV1 and CV2. (3) Gains with ME over SE attributed to genetic correlation between environments, with little effect from residual correlation. Impacts of marker density on predictions also were investigated in this study. Traditional quantitative trait loci (QTL)-based prediction (QP) in marker-assisted selection (MAS) is defined as a two-step process to predict breeding values (BVs) of untested lines for traits of interest based on QTL identified from a typical biparental plant breeding population. In the first step, QTL are identified based on phenotypic and genotypic data via various QTL mapping methods such as stepwise regression , intervaThe accuracy of QP is contingent on the numbers of QTL identified and their respective estimated effects. Although it is relatively easy to detect QTL with large phenotypic effects, it is difficult to identify QTL with intermediate or small effects ; Xu 2003Simulation and empirical studies have shown that genome-wide prediction (GWP)  prediction model in which phenotypic records or means of a training sample and a validation sample are obtained from the same set of environments by using the same year and location effects, with no consideration of genetic and residual correlation across environments. Recent studies have illustrated that the use of genetic and residual covariance across correlated environments may better the accuracy of across-environment GWP using a multienvironment (ME) model (http://www.panzea.org). A total of 1106 single-nucleotide polymorphism (SNP) markers were used for genotyping each RIL, covering a genetic map of 1439 cM. The coding rules used for genotypic data were identical to those used in the within-environment prediction shown in H2 on a mean basis  ranged from 0.60 to 0.84 for the LL trait and from 0.60 to 0.82 for the LW trait (Table S1). Over 25 NAM populations, the average of H2 was 0.74 for both traits, reflecting a high precision of phenotyping efforts.We retrieved phenotype and genotype data of 4131 recombinant inbred lines (RILs) from maize NAM populations derived from crosses between 25 genetically diverse inbreds and the maize elite parent B73  from theTwo methods, CIM and RR-BLUP, were used to estimate marker effects based on genotypic and phenotypic data in a training data set in CV1 with balanced phenotypic records and CV2 with unbalanced phenotypic records which were discussed in details in next section. Prediction accuracies with CIM and RR-BLUP were used to measure estimates with QP and GWP strategies, respectively. Two models, SE and ME, were applied in both methods based on training sets obtained from WP and AP. In total, there were four models developed for each method: AP-ME, AP-SE, WP-ME, and WP-SE. In the discussion of each method, we start with the most complex model, AP-ME, and then simplify to obtain others.Y = T is an n × m matrix of phenotypic data for n lines from p NAM populations and m environments, and y1, y2,Δ,yn are vectors composed of observations across m environments; X is an n × (1 + p) incidence matrix for n lines from p populations representing population structure with the first column ones; β is a (1 + p) × m matrix of individual population effects with the first column representing the overall mean; W is an n × f matrix of genotypes of cofactor markers; s is an f × m matrix of cofactor marker effects with f the number of cofactor markers; Q is an n × 1 matrix of QTL genotypes; α is an 1 × m matrix of additive effects of a putative QTL at a tested position; and ε = T is a matrix of residuals εi , which is assumed to be correlated between environments and to follow a multivariate normal distribution with means zero and covariance matrixβ, cofactor effects s, and QTL additive effects α were treated as fixed effects and residual ε as random effects. QTL genotype Q was not observed and was replaced with its expected value obtained from the probability distribution of QTL genotypes conditional on the closest flanking markers  score assumed its maximum in the region under consideration with a LOD threshold . In WP-Syi  is a vector of phenotypic data for n lines from p NAM populations at the environment i; m is the total number of environments; Xi is an n × (1 + p) incidence matrix for n lines and p populations representing the population structure with the first column ones; βi is a vector of individual population effects ; Zi is an n × k matrix of marker genotypes with k the total number of markers; αi is a vector of marker effects; and εi is a vector of residuals. The above model can be rewritten in reduced matrix form asY = T; β = T; α = T; and εT = T. In this model, the population effects β were treated as fixed effects. The marker effects α were treated as random effects following a multivariate normal distribution G0 the additive genetic covariance matrix between α and α′, IK a k × k identity matrix, and ⊗ the Kronecker product of matrices. G0 can be defined asiiσ2 is the genetic variance of lines in environment i , ijρ  is the genetic correlation of lines between environment i and j; and ijρ = jiρ. The genetic covariance matrix of Y can be defined asA is an n × n genetic relationship matrix, which can be calculated using genome-wide markers asM is an n × k matrix of marker genotypes and ip is the frequency of an allele at locus i   in the CV1 scheme. The residuals ε were considered as random effects following a multivariate normal distribution N where R is a residual covariance matrix calculated byIN is an n × n identity matrix, and R0 is an m × m matrix of the residuals in different environments, written asiie2 is the residual variance in environment i , ijr  is the residual correlation of lines between environment i and j, and ijr = jir. When model (2) was modified into AP-SE, it was reduced to a univariate model, and G0 and R0 became a special case of the multivariate model with diagonal entries denoting the genetic and residual variances, respectively. In WP-SE and WP-ME, the population structure X was reduced to the column vector of ones. These models were applied to estimate marker effects based on training data sets with balanced phenotypic records in CV1. For training sets with unbalanced phenotypic records in CV2, marker genotype matrix Zi in environment i may not be equal to Zj in environment j. The genetic covariance between line l and q across environment i and j in G was modified into Gijlq = {ZilZjqTσij2/2∑hp(1 - hp)} where Zil is the genome-wide marker genotype row vector for line l in environment i, Zjq is the genome-wide marker genotype row vector for line q in environment j, ijσ2 is the entry of G0 at row i and column j, and hp is the frequency of an allele at locus h . The residual covariance was Rijlq = {ije2} with ije2 the element at the ith row and the jth column of R0 if line l in environment i and line q in environment j are from the same genotype, and zero otherwise,The AP-ME model with RR-BLUP was an extension from the model proposed by 2, Δ, k) . Note thG0 and R0. In this study, G0 and R0 were estimated by the multivariate restricted maximum likelihood (REML) approach proposed by G0 and R0 were estimated in each NAM population for WP, they were estimated once by combining full data sets from all NAM populations for AP. Once estimation for G0 and R0 were done, environment-specific marker effects α along with the population effects β were estimated by solving Henderson’s mixed-model equations (A key requirement in model (2) was to estimate the covariance matrix quations .We used validation methods to estimate the prediction accuracy of each model described previoulsy. Cross-validation is a statistical technique of splitting data into training and validation sets by using the validation set to evaluate prediction ability of the trained model. Two cross-validation schemes, CV1 and CV2 , were co4 = 2.56% across four environments. Excluding these lines had little effect on CV2 analysis.In each replicate of CV1 , RILs inIn both CV1 and CV2 described above, training and validation sets were from a single population by the use of WP information. As an alternative practice, AP information can be used for prediction in the absence of WP information  based on identical validation sets within each NAM population. Gains in prediction accuracy with one model  over another one  were calculated using (AR – BR) / BR, where AR represents prediction accuracy with model A and BR prediction accuracy with model B.Although training was performed with WP and AP approaches in CV1 and CV2, validation was conducted within each NAM population for each environment. In each replicate, prediction accuracies were measured as the correlation coefficient between estimated BVs and observed phenotypes of lines in the validation set usingG0 and R0 estimated above were set to zero, imposing independence on genetic and residual correlations. This model was equivalent to fitting each SE model separately with GWP. SG-UR imposed independence on genetic correlation by setting non-diagonal entries to zeros in G0, while UG-SR removed residual correlation by setting non-diagonal entries to zeros in R0. UG-UR was completely unstructured and estimated in the previous section. Influences of residual covariance were evaluated by comparing SG-SR with SG-UR, whereas impacts of genetic covariance were assessed by comparing UG-SR with SG-SR. Finally, effects of residual covariance given the genetic covariance were evaluated by comparing UG-UR with UG-SR.Impacts of genetic and residual covariance on GWP with ME were assessed by comparing four different models: (1) structured genetic and residual covariance (SG-SR); (2) structured genetic and unstructured residual variance (SG-UR); (3) unstructured genetic and structured residual variance (UG-SR); and (4) unstructured genetic and residual covariance (UG-UR). SG-SR was the simplest model among them, in which each of the nondiagonal entries in c) between two flanking markers to ensure that each chromosome was evenly covered by a set of SNPs. For each NAM population, varying numbers of markers were obtained by selecting values of c  without compromising the overall genomic coverage using the method described by c = 1.6 cM, meaning that the mean marker distance of the entire marker set was ~1.6 cM. Note that a training sample proportion 0.6 was always used when assessing the effect of different marker densities to ensure different densities were applied to the same training and validation sets at each replicate of 100 cross-validations.Effects of marker densities on GWP were tested by setting a conditional genetic distance criterion . As discussed previously, QTL for QP were identified using the empirical LOD thresholds estimated by permutation tests (Table S2). With GWP, the genetic and residual covariances obtained from multivariate REML for traits LL and LW  were used to estimate environment-specific marker effects with RR-BLUP. We first compared GWP with QP in CV1 and CV2. Overall, GWP gave consistently greater accuracies than QP. Increase in accuracy with GWP over QP was statistically significant in 99% of predictions generated . Gains in accuracies with GWP over QP were larger for WP than for AP were attributed to two factors: (1) GWP with WP gave greater accuracies than that with AP; and (2) QP with WP gave lower accuracies than that with AP due to a decreased number of QTL identified in WP. Although AP gave improved accuracies for QP over WP, it still yielded lower predictive power than GWP, which was also lower than the accuracies with GWP in WP.Accuracies of predictions for traits LL and LW are shown in Table S7, Table S8, Table S9, Table S10, Table S11, Table S12, Table S13, Table S14, Table S15, Table S16, Table S17, Table S18, Table S19, Table S20, Table S21, and Table S22). For WP in CV1, across all populations and environments, average gains in accuracies with ME over SE were 8% and 12% for traits LL and LW, respectively. Gains attained a maximum of 48% in some populations in CV1 (Table S7). For WP in CV2, average increases with ME over SE reached 40% and 33% for LL and LW traits, respectively. For AP, gains with ME over SE were reduced to 3% in CV1 and 10–13% in CV2 for LL and LW traits. On average, across all populations, environments, and traits, gains with ME over SE were 10% and 37% for WP, and 3% and 11.5% for AP in CV1 and CV2, respectively. The aforementioned gains were observed based on the average genetic correlation of 0.77 and 0.87 and the average residual correlation of 0.31 and 0.39 for WP and AP, respectively . With different training and validation sets, it may not be appropriate to evaluate performances of GWP between CV1 and CV2 on the same ground in this study. However, given similar accuracies with the SE model obtained from CV1 and CV2, comparison between them suggested, but did not prove, that gains with ME over SE were greater in CV2 than that in CV1 at WP and AP.Afterward we compared prediction accuracies between the SE and ME models for GWP. The ME model provided greater prediction accuracies than the SM model in both CV1 and CV2. Increases in accuracies were significant in most cases (Table S23, Table S24, Table S25, Table S26, Table S27, Table S28, Table S29, Table S30, Table S31, Table S32, Table S33, Table S34, Table S35, Table S36, Table S37, and Table S38. In CV1 and CV2, SG-UR gave prediction accuracies comparable to or less than that attained by SG-SR, suggesting that addition of residual covariance did not benefit predictions, and may have worsened them. In contrast, prediction accuracies with UG-SR were greater than that with SG-SR, suggesting that modeling genetic covariance improved predictions with the ME model. Finally, UG-UR gave comparable prediction accuracies to UG-SR, suggesting that adding residual covariance had little effect on predictions in the presence of the genetic covariance.Prediction accuracies for GWP using ME with different genetic and residual variance structures are summarized in Marker densities of 1.6, 5, 10, 15, 20, 25, and 30 cM were used corresponding to different numbers of markers in each training sample via methods described in the previous section. As expected, prediction accuracies for GWP with SE and ME models improved with increasing marker density in the example using the NAM population B73×CML322 . These iGWP provides a significant predictive advantage over QP in WP and AP, with increased gains in WP compared with AP. For almost all cases in WP, GWP shows significantly greater accuracy than QP in each of the NAM populations. This conclusion is consistent with the results from previous studies .The use of across-environment information improves prediction with GWP. Two main results were obtained from this study. First, modeling covariances between correlated environments with the ME model gives better predictions compared with SE in both the CV1 and CV2 schemes. At an average genetic correlation of 0.77, gains of 10% in CV1 and 37% in CV2 were observed for WP, greater than that of 3% in CV1 and 11.5% in CV2 for AP at a correlation of 0.87. Although the superiority of ME over SE in CV2 can be explained by borrowing information from the same lines across environments . FurtherImpacts of marker density also were evaluated for GWP with the SE and ME models. Overall, our conclusions were consistent with previous work from within-environment predictions (In practical plant breeding, GWP needs to be integrated into the appropriate genome-wide selection schemes where several cycles of intercrossing may be required to increase genetic gains . At its