We are now well into the sequencing era of genetic analysis, and methods to investigate rare variants associated with disease remain in high demand. Currently, the more common rare variant analysis methods are burden tests and variance component tests. This report introduces a burden test known as the modified replication based sum statistic and evaluates its performance, and the performance of other common burden and variance component tests under the setting of a small sample size  using the Genetic Analysis Workshop 18 simulated data with complete knowledge of the simulation model. Specifically we look at the variable threshold sum statistic, replication-based sum statistics, the C-alpha, and sequence kernel association test. Using minor allele frequency thresholds of less than 0.05, we find that the modified replication based sum statistic is competitive with all methods and that using 103 individuals leads to all methods being vastly underpowered. Much larger sample sizes are needed to confidently find truly associated genes. We are now well into the sequencing era of genetic analysis, and methods to investigate rare variants associated with disease remain in high demand. The typical methods for detecting single variants associated with disease are not suited for the rare variants, owing to the low minor allele frequency (MAF) of rare variants. Most rare variant analysis methods fall into two categories, burden tests  and vari+) or cases (S-) to accommodate both protective or deleterious rare variants within the same gene ) or sum the weighted sums (Scomb = S- + S+) [- and S+ based on the data. The C-alpha statistic is a variance component method that compares the distribution of the rare variants in the sample with that expected under the null, assuming a binomial distribution of each rare variant [We begin by briefly summarizing some of the more common and more recent methods for rare variant analyses and give their primary references. We consider the following burden tests: the variable threshold sum statistic (VT)  and the ame gene . Then thS- + S+) . In this variant . And fin variant . Both C-Additionally, a major issue faced with researchers investigating rare variants, beyond the low MAF, is sample size considerations. Although some consortia exist, providing larger sample sizes for rare variant analyses, many researchers following up genome-wide association studies (GWAS) do not have the resources to sequence the full GWAS sample, leading to sample sizes on the order of only a few hundred for rare variant analysis. One recent study used 40 individuals in total .As with all methods, larger samples give more power, and limited sample size limits the power. With rare variants, this is compounded because only a few individuals in the sample will exhibit a rare variant. For smaller samples, the MAF of the rare variant must be larger to even have a chance of appearing in the sample. But would a rare variant analysis still be useful for the rare variants that do appear in the sample? And which method would be effective in detecting the rare variants with smaller sample size? In addition to examining our new method, we investigate these questions using the simulated data of the Genetic Analysis Workshop 18 (GAW18)  Because we are comparing the performance of methods on the same data, we focused on genes on chromosome 3 that were used in the simulation model. We used NCBI dbSNP to identify all the single-nucleotide polymorphisms (SNPs) belonging to each gene on chromosome 3 involved in the simulation model.We evaluated type 1 error, based on a resampling approach, specifically, we simulated a dichotomous phenotype to be independent of the genotype following the method in . We simumax, Scomb, Stau, C-alpha, and VT for chromosome 3 for each replicate using the simulated phenotype. We also used the freely available R-package for computing SKAT using the default values with the small sample size option [For the burden tests, we computed a burden statistic for each gene. We computed Se option . For eacThe signal strength for each gene on chromosome 3 ranged from 0% to 4.5% . The actual raw number of causal variants per gene ranged from 0 to 9, and the number of total variants per gene ranged from 5 to 2531. Thus, the total number of rare variants (denominator) controlled the signal to noise ratio . Power exceeded 10% for only 9 of the 30 causal genes: ARHGEF3, FBLN2, GPR160, PAK2, PTPLB (SKAT only), RAD18 (VT only), RYBP (VT only) SERP1, TFDP2, and TUSC2 , the VT method is substantially more powerful than the other methods, but for others , it is less powerful. Regardless of the variance in power across methods across genes, when averaging each method's power across all genes, the averages were close to 0.05  (R2 >0.25) with a causal variant in MAP4 that has a large effect size for both diastolic blood pressure (DBP) and systolic blood pressure (SBP). Because this is a gene detected by all methods with power greater than 0.2, the power could be driven through the LD with the large effect size combined with the small number of total variants included in the weighted sum. Beyond gene size and LD, using a dichotomous hypertension variable rather than the continuous trait could have cost power and added complexity to determine the efficacies of these methods with such a small sample size versus using SBP or DBP as a continuous trait.In fact, all methods investigated here are underpowered for this case-control study with a sample size of 103 individuals. These methods fail to detect the causal rare variants most of the time . It is worth further investigation to more clearly determine any advantages for using Stau.The main conclusion of this paper is that for all methods, 103 individuals are not enough for a rare variant analysis of a complex qualitative disease such as hypertension. Perhaps power can be increased using blood pressure as a continuous trait rather than treating hypertension status as a qualitative trait. A secondary conclusion from this report is that the new method SThe authors declare that they have no competing interests.tau, performed some analyses, and wrote the manuscript. TK refined Stau; wrote computer code to compute Stau, Smax, Scomb, and a script to perform all analyses; and provided editorial input on the manuscript. JN wrote computer code to compute VT and C-alpha and provided editorial input on the manuscript. RKY performed data cleaning and preliminary analyses and provided editorial input on the manuscript. SS consulted on the analysis and interpretation of results and provided editorial input on the manuscript II-L provided consultation for the computer code for Stau, Smax, and Scomb; analysis and interpretation of results; and editorial input on the manuscript. All authors read and approved the final manuscript.MDS designed the overall study, conceived of S


Among various previously proposed methods to estimate heritability, variance component methods are advantageous when estimating heritability using markers. Due to the high-dimensional nature of data obtained from genome-wide association studies (GWAS) in which genetic architecture is often unknown, the most appropriate heritability estimator model is often unclear. The Haseman–Elston (HE) regression is a variance component method that was initially only proposed for linkage studies. However, this study presents a theoretical basis for a modified HE that models linkage disequilibrium for a quantitative trait, and consequently can be used for GWAS. After replacing identical by descent (IBD) scores with identity by state (IBS) scores, we applied the IBS-based HE regression to single-marker association studies (scenario I) and estimated the variance component using multiple markers (scenario II). In scenario II, we discuss the circumstances in which the HE regression and the mixed linear model are equivalent; the disparity between these two methods is observed when a covariance component exists for the additive variance. When we extended the IBS-based HE regression to case-control studies in a subsequent simulation study, we found that it provided a nearly unbiased estimate of heritability, more precise than that estimated via the mixed linear model. Thus, for the case-control scenario, the HE regression is preferable. GEnetic Analysis Repository (GEAR;  M) is far greater than the number of individuals (N), estimating heritability is difficult. For instance, if the statistical power is insufficient, variants associated with a small effect may not be captured under a stringent p-value threshold (~ 10−8). This obstacle can be partially bypassed by implementing the mixed linear model, which uses the genetic relationship between individuals estimated from single nucleotide polymorphism (SNP) markers in lieu of fitting hundreds of thousands of markers together  regression is a prestigious method for estimating variance components  times faster than the mixed linear model. In order to facilitate the application, the HE regression algorithm for GWAS data was implemented in Java software, GEnetic Analysis Repository (GEAR), which is freely available online.Furthermore, as the IBS-based HE regression uses least squares, it is advantageous in its computational efficiency and is As the first half of this report is focused on establishing the mathematical basis of the IBS-based HE regression, many mathematical symbols are introduced Table . In the yi, which follows the normal distribution of N, and the genotype is xi = , in which M is the number of markers. For the ith individual, the genotype at the kth locus is xik, which counts the reference alleles at the kth locus. The reference allele is denoted as Ak and the alternative is ak. The frequency of Ak is pk and the frequency of ak is qk. gk is the set of possible genotypes, say {kak, Akak, AkAka}, at the kth locus. Consequently, akak, Akak, and AkAk are coded as 0, 1, and 2, respectively. After standardization, xi is expressed as xik, given a genotype of akak, Akak, and AkAk, their standardized scores are lth QTL is denoted as βl. Throughout the study, we assume a polygenic model with L QTLs.For an individual, the phenotype is denoted as Yij = μ + bπij + eij, for detecting linkage between a marker and a QTL in a full-sib design. Yij represents the squared difference between a pair of full sibs, and πij is the proportion of IBD at an observed marker locus; μ is the intercept of the regression, b is the regression coefficient, and eij is the residual. The mathematical expectation of the regression coefficient is b = −2(1 − 2c)2 σ2A, in which c is the recombination fraction between the marker locus and the QTL, and σ2A is the additive genetic variance of the QTL.Haseman and Elston  proposedN unrelated individuals. If the phenotype for the ith individual is yi, we can modify the HE original regression as belowYij = (yi − yj)2 represents the squared difference, Ωij is the measure of the genetic relatedness of a pair of individuals, and eij is the residual. Given N unrelated individuals, there are  such individual pairs. Ωij is the similarity score between a pair of individuals based on the IBS, as recently proposed  is the variance of the genetic relatedness. cov = E(ΩijYij) − E(Ωij)E(Yij) = E(ΩijYij) because E(Ωij) = 0 [see the definition for Ωij in section The Derivation of var(Ωij) and Effective Number of Markers (Me)], and ij and Yij. In order to derive var(Ωij) and cov, we need to introduce the haplotype distribution of a biallelic loci pair . When the haplotype is constructed on a pair of markers, it leads to the derivation of var(Ωij) [section The Derivation of var(Ωij) and Effective Number of Markers (Me)]; when the haplotype is constructed for a marker and a QTL, it leads to E(yi|xik), the conditional expectation of the phenotype based on a marker [section The Derivation of E(iy|ikx)].For the linear model in Equation (1), the expectation of the regression coefficient is klr = p(la|ka) and klR = p(lA|kA) are defined as the conditional probabilities of the haplotypes in the coupling phases, such as kala and kAlA, respectively; 1 − klr and 1 − klR represent the conditional probabilities of the alleles in their repulsion phases, such as kAla and kalA, respectively. klD = kAlAf − kplp, in which kAlAf is the frequency of the haplotype kAlA; klD is the covariance between the loci, quantifying the LD between them.For a pair of biallelic loci, there are four haplotype phases, and their conditional probabilities are as summarized in Table 2kl is often used to parameterize the LD of a loci pair p(A = kAk|AlAl)p(AkAk)p(A = l|Ak)p(Ak) p(Al|Ak)p(Ak)p = 0 and cov = E(ΩijkΩijl) − E(Ωijk)E(Ωijk) = E(ΩijkΩijl) because E(Ωij.) = 0.The possible relatedness scores of a pair of individuals are summarized in Table ij is informative in revealing hidden relatedness. For example, for the duplicated individual in the sample, E(Ωij) = 1; for first-degree relatives, E(Ωij) = 0.5; for second-degree relatives, E(Ωij) = 0.25. Consequently, it can control the entry of samples that are under the expected cutoff for relatedness.ΩAfter some additional algebra  = 0; when the kth locus and the lth locus are at the same locus, cov = 1.When the 2kl varies with kp and lp  as the mean of the squared Pearson's correlation between the markers along the genome, denoted as ρ2M.The distribution of ρpl Wray, . We can eM, is introduced here. Intuitively, as markers are often in linkage disequilibrium, the real number of “independent” markers is smaller than the total number of the markers genotyped. This concept was previously introduced under the context of risk prediction  is the mathematical expectation of the effective number of markers evaluated under the simulation method  = Σilglxsilp(xil|xik = kAkA), in which lg = {ll, llq, lqlq}, and E(iy|ikx = kAkA) = βl × klR2 + 0 × 2klR (1 − klR) − βl × (1 − klR)2 = (2klR − 1)βl. Analogously, we can derive the expected values of E(i|xiky = kakA) = (klR − klr)βl and E(i|xiky = kaka) = (1 − 2klr)βl (See Table E(i|xiky) is defined, the distribution of E can be tabulated as in Table The expected phenotype of M markers, each of which can be in LD with L QTLs.In this section, we investigated two scenarios to derive the expected value of the regression coefficient for Equation (1). In scenario I, genetic similarity is estimated at a single marker, which is in LD with one or more QTLs. In scenario II, genetic similarity is estimated based on kth marker, and one QTL, say the lth QTL, since var(Ωijk) = 1, E(b) = E(ΩijYij), which is E(ΩijYij) = ΣikxΣjkxsiksjk[i|xik)E(y − j|xjk)E(y]2p(ikx)jk)p = −4kqkp β2l because rkl = klR = 1. When the QTL is in linkage equilibrium with the marker, rkl = lq and Rkl = lp, 1 − klr − klR = 0, and consequently E(b) = 0.When the QTL overlaps with the marker, or the correlation between the QTL and the marker is 1, E(b) = − 2ρ2klσ2l, in which According to Table in lieu of the recombination fraction for linkage. The mathematical expectation of the regression coefficient resembles the one in the original HE regression. However, it should be noted that here the interpretation of the regression coefficient is based on linkage disequilibrium and association, whereas the original interpretation is based on linkage between the marker and the QTL.In the GWAS context, squared LD (Pearson's correlation) is yi given xik is ij and ijY is as summarized in Table E(ΩijYij) = ΣikxΣjkxsiksjk[E(yi|xik) − j|xjk)E(y]2ik)p(xjk)p can be rearranged asL = 1, Equation (9) can be simplified to Equation (7).It is easy to see that when M markers, each of which may be in LD with L QTLs, the HE regression becomes Yij = a + bΩij, in which ijk denotes the relatedness fraction constructed with the kth marker between the ith and the jth individuals. According to the definition of the regression coefficient, When the genetic relatedness matrix is constructed with cov = ρkl2, as expressed in Equation (4). and  is the average squared LD between a marker and a QTL across the genome, and After rearrangementIf the phenotype is standardized, heritability equals the additive variance component. It is straightforward to obtain an estimate of the heritability for a single QTL, as in scenario I, or all QTLs, as in scenario II (See Supplementary Note var(ijY) = 8σ4y .The sum of square error (SSE) isvar(Ωij) = 1 and ρ2M = 1.For scenario I, as only one marker is used,  and 8Me » 4σ4A Λ2.Given the current GWAS data, which incorporates thousands of individuals and often up to one million markers, it is reasonable to assume For real GWAS data with about one million markers, When the phenotype is standardized, the sampling variance of the regression coefficient is half of the additive variance component.E(ijY) = E[(iy − jy)2] = E(iy2) + E(jy2) − 2E(iyjy). E(iy2) = var(iy) − E(iy)2, E(jy2) = var(jy) − E(jy)2, E(iyjy) = cov + E(iy)E(jy). As the individuals are not related to each other, assuming no common environment, cov = 0. So, E(ijY) = var(iy) + var(jy) = 2σ2A + 2σ2e, twice the phenotypic variance. The negative ratio between the regression coefficient and the intercept provides an estimate of the heritability if the phenotype is not standardized.The expectation of the intercept is The derived regression coefficients and their sampling variance at the completion of the derivation are summarized in Table h2Λ.The additive variance of a trait is defined as In fact, the HE regression and the mixed model are equivalent and can agree on the heritability estimate (see Simulation III). However, this equivalence can be disturbed when QTL effects are not randomly distributed (Simulation IV).h2l is the heritability on the liability scale, h2o is the heritability on the observed scale directly estimated based on the case-control data, K is the prevalence of the disease, P is the proportion of the cases in the data, and z is the height of the standard normal distribution in which the prevalence K is located.Like the debut application of the original HE regression for schizophrenia .Y in the HE regression can also be expressed as a cross-product, and then E(b) = −2kqkpτ2klβ2l, which is half that of Equation (7)  for a single-marker application. One thousand unrelated individuals were simulated. One marker and one QTL were simulated, both of which were equifrequent and biallelic. The heritability of the QTL was 0.5. The LD between the marker and the QTL was set at three levels: ρ = 0.25, ρ = 0.5, and ρ = 0.75. The single marker was used to construct the genetic relatedness, Ω. Then a single-marker-based HE regression was conducted. After standardizing the phenotype, the negative half of the regression coefficient returned the unbiased heritability estimate.As indicated by Equation 7), given ρ = 0.25, ρ = 0.5, and ρ = 0.75, the regression coefficient expectation was −0.062, 0.125, and 0.57, respectively. After 100 rounds of simulation, the derived expectation of the regression coefficient, as well as the sampling variance  and the required sample size to detect a QTL are indicated.For the single-marker HE regression, as the expectation and the sampling variance of the regression coefficient were already derived, a yi = μ + ksikb + ie, if the phenotype and the genotypes are both standardized, E(kb) = ρklβl, and its standard error is t-test can be constructed as t statistic, the non-centrality parameter of In contrast, for a conventional one-marker association linear regression, 2 tests differ by the factor kl = 0.25), medium (ρkl = 0.5), or high (ρkl = 0.75), the sample size required to allow HE to outperform the linear regression is 6400, 1600, and 712, respectively. If the heritability is even smaller, say h2 = 0.001, the required sample size is 12,800, 3200, and 1423 to make the HE regression more powerful under the low, medium, and high LD, respectively.These two χDepending on the sample size, heritability, and LD patterns between the QTL and the target marker, the power of the HE regression may or may not be greater than the conventional linear regression. However, when the sample size is large, or the heritability of the QTL is large, HE regression is a more powerful tool for association studies. These results are based on the assumption that the real sampling variance agrees with the derived theoretical result.N. Four LD levels  were adopted for each of two consecutive QTLs, and the effective number of markers decreased correspondingly . One thousand unrelated individuals were simulated, and the genetic relatedness of each pair of individuals was estimated on these 100 QTLs. The heritability of the simulated polygenic model was 0.5, which is calculated as In this simulation, 100 equifrequent and biallelic QTLs were simulated, and the additive effect of each QTL was sampled from yi = μ + ijajx + ie, where yi is the phenotype of the ith individual, μ is the mean, xij is the indicator variable with values of 0, 1, or 2 depending on the reference allele counts, and ej is the residual. Restricted maximum likelihood (REML) was employed to estimate the variance components of the mixed linear model  = −2σ2A Λ (ignoring Δ) sheds light on the inference of the general LD pattern between the tagged markers and the causal variance. ρ2M can be estimated from markers. If the heritability of the trait is known (not likely though), it is possible to estimate ρ2Q. For example, the heritability of height is estimated at around 0.8  still could predict the results of the HE regression correctly , and σ2b = lh2/[2 × p × (1 − p) × L], in which p = 0.5. The phenotype of each individual under the liability scale was scaled to unit. The ascertainment of cases on the liability scale was K. Individuals were sampled from the described reference population until 1000 cases and 1000 controls were recruited.The HE regression was applied to case-control data. A polygenic model of L = 100, 1000, and 10,000, and three levels of disease prevalence at the population level, K = 0.1, 0.01, and 0.001, were adopted. Nine scenarios were simulated in total, and 30 independent simulation replications were implemented for each scenario.The heritability on the liability scale was 0.5. In order to cover a broad range of scenarios, three levels of QTL number, The genetic relationship matrix was constructed using all individuals and the allele frequencies were estimated from the sample. The genetic additive variance components were estimated with the HE regression and the mixed model method. As the directly estimated variance component was on the observed scale and could be greater than 1, we employed both the REML and non-constrained REML for mixed model methods, which allowed the heritability to be greater than 1.h2l was compared across all three methods. In general the HE regression resulted in a more precise estimate than that of the REML and non-constrained REML. For the mixed model methods, either with or without constraints, REML often underestimated the variance components. The bias was caused by two factors: the number of QTLs (in each row panel) and the prevalence of the disease (in each column panel). With fewer QTLs, a lower prevalence could exacerbate underestimation by the mixed model.As illustrated in Figure The analytical results summarized in Table In order to facilitate application of the HE regression method to estimate complex trait heritability, GEAR software was developed. GEAR was developed on Java and can run across many operating systems, such as Windows, Mac, and Linus/Unix, as long as a Java virtual machine is available. GEAR has been demonstrated to function in the following situations.It can generate genetic relatedness of unrelated individuals, as formulated in Equation (3), based on whole-genome markers.It can estimate the effective number of markers based on a genetic-relatedness matrix.It can estimate heritability with the HE regression. GEAR can read genotype data saved in PLINK binary format  = − 2ρ2 σ2A. As these two models are based on different genetic mechanisms, the interpretations of their respective regression coefficients are reasonably different. In the IBD-based HE regression, E(b) = −2(1 − 2c)2 σlA2, 1 − 2c ranges from 0 to 1; whereas in the IBS-based HE regression, E(b) = −2τ2kl σlA2, in which τkl, ranges from −1 to 1. As the values of r and R rely upon the allele frequencies of the biallelic marker and the biallelic QTL, they reach either −1 or 1 only given that the marker has the same allele frequency as that of the QTL. However, after taking the square, both (1 − 2c)2 and τ2 lie between 0 and 1, inclusive.Under various scenarios, the mathematical expectations of the regression coefficients, as well as the sampling variances, were derived. There is substantial mathematical similarity between the IBD HE regression and the IBS HE regression. For example, for both models under the single-marker scenario, their regression coefficients can be expressed in a unified form, E(b) = −2σ2A Λ (ignoring Δ), provides a possible way to estimate the LD pattern between causal loci and markers. If the true heritability is not readily known, it is possible to estimate ρ2Q, the average LD between QTLs and markers. As demonstrated in simulation III, it may be possible to estimate ρ2. However, the causal loci can be in any possible form, such as SNPs, chromatin markers, or methylation markers, and different methods capture genetic variation in different forms. In practice, the obstacle in estimating ρ2 lies in how heritability estimated from different methods, such as linkage and association, or genotyping platforms, such as SNP markers and methylation markers, can be connected to each other. Equation (11) sheds light on the investigation for how QTLs are distributed along the genome.Equation (11), Application of the HE regression to heritability estimation of complex traits revealed that the HE regression seems to be equivalent to the mixed model approaches in general (Δ = 0). A similar equivalence was previously established for linkage analysis , which indicates that it is cubic to the sample size and t rounds of iterations. The time complex of the HE regression is far lower, asymptotically O(2N2), given two parameters, the intercept and the regression coefficient, included in the model. Given the large sample sizes often employed in GWAS, the computational burden can be dramatically reduced. Although the HE regression method is derived on a simple-regression scenario, its extension to a multiple-regression scenario is straightforward. For instance, the genetic relatedness between each pair of individuals can be constructed on each chromosome and then all chromosome-based relatedness scores can be fit into the regression framework. In addition, the difference between a pair of phenotypes can also be expressed as a cross product and squared sum  software. Guo-Bo Chen wrote the manuscript.The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


Genome-wide association studies have proven successful but they remain underpowered for detecting variants of weaker effect. Alternative methods propose to test for association by using an aggregate score that combines the effects of the most associated variants. The set of variants that are to be aggregated may come from either of two modeling approaches: single-marker or multi-marker. The goal of this paper is to evaluate this alternative strategy by using sets of single-nucleotide polymorphisms identified by the two modeling approaches in the simulated pedigree data set provided for the Genetic Analysis Workshop 18. We focused on quantitative traits association analysis of diastolic blood pressure and of Q1, which served to control the statistical significance of our results. We carried out all analyses with knowledge of the underlying simulation model. We found that the probability to replicate association with the aggregate score depends on the single-nucleotide polymorphism set size and, for smaller sets (≤100), on the modeling approach. Nonetheless, assessing the statistical significance of these results in this data set was challenging, likely because of linkage because we are analyzing pedigree data, and also because the genotypes were the same across the replicates. Further methods need to be developed to facilitate the application of this alternative strategy in pedigree data. As such, for many complex traits, only a few SNPs have been identified, leaving a large part of the trait heritability unexplained. This is partially because a wide spectrum of effects may be implicated, many of which are not detected at the stringent significance criteria levels set in GWAS. One way to circumvent this limitation has been through using larger sample sizes, thus increasing the power to detect SNPs of weaker effect. Following their successful application in a few studies .) For the BLUP strategy, the peak is reached at p values ≥1 × 10−3 and for larger set sizes (S ≥1000) where they are nearly always at 100%. For smaller set sizes (S ≤100), replication rates are greater under the single-marker strategy than under the BLUP strategy. The opposite trend is observed for larger set sizes (S ≥5000).Figure p values <1%). Estimates of the replication rates were close to the theoretical values, whether the top SNPs were selected under the single-marker approach or the BLUP approach, and irrespective of set size S . PS association analyses conducted on the permuted DBP_1 trait yielded to slightly inflated rates (results not shown), especially for larger set sizes S and under the BLUP strategy . From these results it is not clear whether the distribution of the PS association test appropriately follows the theoretical distribution in the pedigree data set even though the traits were decorrelated.To evaluate the significance of these findings, we performed the PS association analyses on Q1. Note that because of the small number of available replicates (199), replication rates could not be estimated at stringent criteria levels  were considered. Our study presents some weaknesses and limitations. PS analysis was conducted using linear regression on the decorrelated traits. Thus, either the way we constructed the sets of independent top SNPs or the fact that the genotypes were the same in all replicates may have led to biased and inflated estimates of power rates. Type I error rates were found close to the theoretical values when analyzing Q1, but not when analyzing the permuted DBP trait. It appears, therefore, that linkage may have affected our PS analyses even if we worked with decorrelated traits. These results suggest that the presently available methods need to be extended to address the challenges of PS association analyses in pedigree data.Using the measured genotype test, a classical approach to detect association in family data, we found that, with the exception of the SNPs in the The authors declare that they have no competing interests.MM designed the overall study; AL contributed to the design and implemented the software for the BLUP model; NB, MS, and MM conducted statistical analyses and drafted the manuscript. All authors read and approved the final manuscript.


This study examines genomic prediction within 8416 Mexican landrace accessions and 2403 Iranian landrace accessions stored in gene banks. The Mexican and Iranian collections were evaluated in separate field trials, including an optimum environment for several traits, and in two separate environments  for the highly heritable traits, days to heading (DTH), and days to maturity (DTM). Analyses accounting and not accounting for population structure were performed. Genomic prediction models include genotype × environment interaction (G × E). Two alternative prediction strategies were studied: (1) random cross-validation of the data in 20% training (TRN) and 80% testing (TST) (TRN20-TST80) sets, and (2) two types of core sets, “diversity” and “prediction”, including 10% and 20%, respectively, of the total collections. Accounting for population structure decreased prediction accuracy by 15–20% as compared to prediction accuracy obtained when not accounting for population structure. Accounting for population structure gave prediction accuracies for traits evaluated in one environment for TRN20-TST80 that ranged from 0.407 to 0.677 for Mexican landraces, and from 0.166 to 0.662 for Iranian landraces. Prediction accuracy of the 20% diversity core set was similar to accuracies obtained for TRN20-TST80, ranging from 0.412 to 0.654 for Mexican landraces, and from 0.182 to 0.647 for Iranian landraces. The predictive core set gave similar prediction accuracy as the diversity core set for Mexican collections, but slightly lower for Iranian collections. Prediction accuracy when incorporating G × E for DTH and DTM for Mexican landraces for TRN20-TST80 was around 0.60, which is greater than without the G × E term. For Iranian landraces, accuracies were 0.55 for the G × E model with TRN20-TST80. Results show promising prediction accuracies for potential use in germplasm enhancement and rapid introgression of exotic germplasm into elite materials. Gene banks are the repositories of novel and useful genetic variation contained in a crop’s gene pool. The wheat gene bank of the International Maize and Wheat Improvement Center (CIMMYT) has access to various germplasm pools, and preserves an array of wild relatives, landraces, genetic stocks, and cultivar germplasm materials. Wheat landrace germplasm has the advantage over wild relatives of being more easily crossed with cultivated hexaploid wheat. The potential of these landraces should be harnessed for present and future wheat genetic improvement programs, and such efforts have begun , which is becoming a standard approach to achieve genetic progress in plants. GS models often accurately predict the value of nonphenotyped plants, and reduce generation intervals by reducing the need for field testing progeny , and days to maturity (DTM) evaluated in drought and heat environments], and several other traits were measured in a single optimum environment; and (2) to study two genomic prediction strategies: (i) random cross-validation schemes where 20% of the accessions form the training (TRN20) set, and 80% of the accessions comprise the testing set (TST80) (TRN20-TST80); (ii) studying the prediction accuracy of two types of reference core sets (diversity and prediction) that included 10% and 20%, respectively, of the total collections to predict the remaining 90% and 80% of the accessions, respectively.In light of the above, the objectives of this study were: (1) to examine the genomic prediction accuracy within a large number of Mexican and Iranian wheat landraces held in CIMMYT’s gene bank for several phenotypic traits; two highly heritable traits were measured in two environments [For traits measured in a single optimum environment, we used the standard Genomic Prediction Best Linear Unbiased Predictor (GBLUP) model for the TRN20-TST80 partitions and the 10% and 20% diversity and prediction core sets. For the two traits that were measured in two environments, we used a G × E reaction norm model (Note that this study focuses on each collection of accessions (Mexican and Iranian separately), and that prediction results from both collections cannot be compared due to differences in population size, number of markers, and traits measured and evaluated in different field experiments. We included results of both collections in the same tables and/or figures for easy presentation, but not for comparing Mexican and Iranian landrace collections.Triticum aestivum) landrace accessions held in CIMMYT’s wheat gene bank. All of the landraces were genotyped using genotyping by sequencing (GBS) methods, and more than 40,000 SNPs were detected. The final number of markers used for each trait in both sets of landraces varied  were planted in different field experiments. DTH and DTM of the Mexican and Iranian wheat landraces were evaluated in field drought (D) and heat (H) experiments at CIMMYT’s experiment station near Ciudad Obregon, Sonora, northwest Mexico , during the 2010–2011 Obregon cycle. Heat stress trials were planted in April 2011, and drought stress trials were planted in November 2010. Both heat and drought are common in wheat growing areas, and they considerably affect several important traits. The drought trial was sown during the normal wheat planting cycle and harvested in April. Plots received ∼250 mm of water through drip irrigation during the entire cropping cycle. The heat stress trial was sown on a delayed sowing date in early April , and harvested in July 2011. This trial was fully irrigated to keep water limitations from confounding the heat-stress results. The heat-stress trial had an average daily maximum temperature of 36.3° (min 18.1°), compared to 27.8° (min 9.0°) for the drought stress trial.2. For each accession of each collection, DTH was determined as the number of days from emergence to 50% spike emergence. DTM was measured as the number of days when 50% of peduncles were completely yellow.In both drought and heat stress trials, fertilization, as well as weed, disease, and pest control, were applied as necessary to minimize limitations. The Mexican and Iranian landraces were evaluated in two separate trials using an augmented grid-check field design with three randomized checks distributed along rows and columns. Plots were 0.40 mOther traits were measured on Mexican and Iranian landraces, but only in a single, optimum, well-irrigated environment in Cd. Obregon using an augmented grid-check field design. For Mexican accessions evaluated in the optimum environment, the other traits measured were: thousand-kernel weight (TKW), test weight (TW), grain hardness (GH), grain protein (GP), SDS sedimentation (SDS), grain yield per square meter (GYSM) and plant height (PHT) . Traits In agricultural field trials, accounting for major and minor sources of spatial variation in plot errors is of paramount importance (PstI and HpaII) were used to create genomic representations of both populations. The samples were submitted to digestion and ligation of barcode adaptors, which allow multiplexing 96 samples in a single lane of an Illumina Hiseq2500 . More than 2,000,000 tags per sample generated up to 77 bases. A DArT P/L analytical pipeline was used to generate allele calls for SNP and silico-DArT. From a total of 40,000 markers, a set of filtering parameters was applied to select and provide high quality markers. Marker filtering was done in a trait-specific manner because some traits were not measured in all the lines; thus these lines had to be deleted from the prediction.The DArT-Seq (h2) , and a second set containing the testing sets on which the prediction model is validated. The authors proposed approximating the PEV as the objective function to be minimized by applying an efficient method that uses the first 100 principal components of the marker data. In this study, the proposed method to approximate the PEV was used to form 10% and 20% prediction core sets that will be the training populations of Mexican and Iranian landraces used to predict the remaining accessions.When sample size increases, computation of that reliability measure is difficult to do in practice. A solution to this problem was recently proposed by We used the standard GBLUP genomic prediction model for traits evaluated in a single environment, and a G × E model that extended the GBLUP theory to a reaction norm model (Genomic accuracy is affected by population structure and/or linkage disequilibrium (LD) between markers and QTL , there is a population structure. Although it is difficult to decide whether to adjust for population structure, we performed the prediction (i) accounting for population structure, and (ii) not accounting for population structure. To account for population structure, analyses were based on the relative contribution to molecular variance. The plots of the proportion of total variance explained by the eigenvalues of the genomic relationship matrix Appendix.When accounting for population structure by correcting the phenotypic response variables of all the traits by the first five principal components, prediction accuracies decreased by about 15–20% (on average) as compared with analyses that did not account for population structure. Full results of the prediction accuracy of models fitted without correcting for population structure are given in the i indexes wheat landraces accessions), on markers using a linear model of the form i = 1,2,…,n accessions; k = 1,2,…,p markers) or, in matrix notation,G given by thi accession is counted in i.e., subtracting i.e., dividing by G converge to twice the coefficient of parentage (or coancestry) between lines.We applied a single-environment model for several traits that were measured in one environment. This model regresses the phenotype vector containing the records for the response variable, UP model , 2008 arResponse variables (DTH and DTM) measured in two environments (D and H) were analyzed by applying a sequence of multiplicative reaction norm models similar to that used by jth accession in the ith environment  and G is the genomic relationship matrix and  are correlated such that model (4) allows borrowing information across The other main effect model adds to model (equation 3) the random effect of the In this study, a Gaussian process with a covariance function structured based on a reaction norm model is used to model the interaction between markers and environments. jth accession in the ith environments  and 80% of the accessions in the testing set (TST80).For traits DTH and DTM measured in two environments (D and H) using the G × E model, we followed The other prediction assessment problems we studied were those posed by using 10% and 20% reference core sets as training populations to predict the remaining 90% and 80% of the Mexican and Iranian landraces, respectively, for the core sets based on diversity or core sets based on prediction. For these prediction problems, the correlation reported for traits DTH and DTH measured in D and H environments is that obtained between the predicted values from the four G × E models, and the observed DTH and DTM values in D and H environments and across both environments. For traits measured in a single environment, the reported correlation is that obtained between the predicted values computed using GBLUP trained with 10% and 20% of the two types of reference core sets and the observed values for 90% and 80% of Mexican and Iranian landraces.The models described above were all fitted using the BGLR R-package . This pahttp://genomics.cimmyt.org/mexican_iranian/traverse/. The Iranian directory contains the following data sets: G.RData, dth_dtm_cores.RDta, standarizedData_dth_dtm.RData, and the standarizedData_univariate.RData. The Mexican directory has one root directory, Toshare, that includes two sub-directories, G×E.Data and Univariate; the G×EData sub-directory has two files, standarizedData_dth_dtm.RData and standarizedData_core.RData, whereas the Univariate sub-directory has two files: Cores and data.Univariate.The complete phenotypic and genotypic data sets for Mexican and Iranian landraces can be downloaded from the link: Appendix).Genomic heritabilities of the traits  to low for PHT (0.116) . The preAppendix) in the TRN20-TST80, 10%, and 20% diversity and prediction core subsets for single environment traits. Prediction accuracies increased up to 0.767 for TKW in Mexican landraces, and up to 0.688 for GW. In general, the 10% and 20% diversity core sets achieved prediction accuracies similar to those of the TRN20-TST80 random partition. However, the 10% and 20% prediction cores did not outperform the diversity core sets or the cross-validation TRN20-TST80.The correlations increased when not accounting for population structure (Box-plots of traits DTH and DTM in D and H environments are depicted in vs. DTH-H) to 0.8276 (for DTH-H vs. DTM-H) (data not shown). For Iranian landraces, sample phenotypic correlations among the four trait-environment combinations are lower than those obtained for Mexican landraces, ranging from 0.3459 (for DTH-D vs. DTM-H) to 0.7939 (for DTH-D vs. DTM-D) (data not shown).Sample phenotypic correlations among the four trait-environment combinations for Mexican and Iranian landraces show that the correlations among traits  for Mexican landraces are high, ranging from 0.5919  random main effect, the G × E variances, and the residual variance explained the largest proportion of DTH and DTM variances within environments. The G main effect explained up to 28.15% (M4) of the variance for trait DTH, and up to 37.38% (M2) for trait DTM . The resThe estimated variance due to accession (A) for all traits and models was much lower than the variance associated with G, suggesting that genomic markers are able to capture a sizable proportion of the total variability due to the main effects of accessions. Estimates from models M3 and M4 for DTH show that roughly 26–28% of within-environment variability can be explained by the main effects of markers (G), 24–26% by G × E interaction terms, and 25–34% by residuals (unaccounted factors). For DTM, the variance components of models M3 and M4 show that 33.65–35.01% of within-environment variability can be explained by main effects of markers, 19.58–20.96% by G × E, and 19.63–26.56% by residuals . For botThe magnitudes of the G variance component for models M3 and M4 were similar, and explained 19.94% (model M3) of the within-environment variance for DTH and 31.65% (M3) for DTM. The G × E and residual variances also explained a large proportion of the within-environment variances for both traits . These rEstimates from model M3 for trait DTH in Iranian landraces indicate that roughly 20% of the within-environment variability can be explained by main effects of markers (G), and 42.77% by interaction terms (G × E), with a residual explaining 28.23%. For trait DTM, the variance components of model M3 indicated that 31.65% of the within-environment variability can be explained by main effects of markers, 30% by G × E interaction terms, and 24.71% by residuals . SimilarAppendix). The variance components do not change much when not accounting for population structure vs. variance component estimates when accounting for population structure . This was expected due to the small variance component of A and A × E as compared with the variances of G and G × E (Appendix).The average GS accuracy of the four models for TRS20-TSN80 for both traits clearly shows that models M3 and M4 gave the highest accuracy across both environments , where pnd G × E . When noAppendix).The average GS accuracy of the four models for cross-validations TRN20-TST80 for both traits across environments are reported in Models M3 and M4 gave the best prediction accuracy across environments for DTH and DTM for the 10% and 20% diversity and prediction reference core sets. When the 10% and 20% diversity and prediction core sets of Mexican landraces were used to predict the remaining 90% and 80% of the Mexican accessions, correlations using models M3 and M4 for DTH and DTM ranged from 0.555 (10% diversity core DTH) to 0.584 (20% prediction core DTM) . The accAppendix).On average across environments, there is a slight decrease in prediction accuracy (about 4%) when the size of the core decreases from 20 to 10%. Also, the prediction accuracy of the 20% diversity and prediction core sets for M3 and M4 (0.58) were similar to the prediction accuracies obtained from random cross-validation TRN20-TST80 across environments (around 0.596). These results confirmed that the 20% diverse and prediction core sizes produced good accuracy when predicting the remaining 80% of Mexican landraces. For the Mexican landraces, a clear increase in prediction accuracy of models M3 and M4 was achieved when not accounting for population structure, with prediction accuracies ranging from 0.717 to 0.758 for both sizes of diversity and prediction cores  and the 10% and 20% predictive core sets were as good as the 10% and 20% diversity core sets at predicting the testing set. Also, the differences in prediction accuracy between the 10% and 20% diversity and prediction core sets were small.The prediction accuracies of the 20% diversity core set of Iranian landraces observed in In this research, we studied genome-enabled prediction accuracy in two different collections of gene bank accessions, one Mexican and one Iranian. Prediction accuracy obtained without accounting for population structure was always higher (15–20%) than prediction accuracy obtained when accounting for population structure. However, in both cases, the prediction accuracy for highly heritable traits in wheat gene bank accessions ranged from intermediate to high. These results may stimulate the application of genomic prediction in gene bank evaluation and germplasm enhancement programs to speed up the process of introgressing diversity into elite germplasm. In general, results of this study are in agreement with the simulation study by vs. TRN20-TST80 was greater (10.3% and 14.9%) than the percent change in accuracy of the 20% diversity for Mexican and Iranian collections (2.7% and 2.8%). The percent change in accuracy of the 20% prediction core vs. TRN20-TST80 was lower than the percent change in accuracy of the 20% diversity core for the Mexican collection.The TRN20:TRT80 analyses represent a random selection of lines for the training population. This set always produced higher accuracy than did the diversity and prediction core sets . For travs. 7.5 7.6 vs. 12.1 5.11 vs. 8.30 1.2 vs. 6.5).For traits measured in two environments (DTH and DTM), in four cases, the 10% and 20% prediction cores had a smaller reduction in their correlations with respect to the TRN20-TST80 set, than those observed for the 10% and 20% diversity core sets for the Mexican and Iranian collections  , while the core sets focus directly on the subset that will produce high correlations between training-testing sets.The main advantage of the 10% and 20% diversity and prediction core sets over the random cross-validation is that they generate in one time a good prediction training set. The random cross-validation partitions explore and sample the entire correlation space between observed and predicted values (n > 1600), while the 20% core sets of the Iranian landraces were relatively small (n > 400). However, genomic prediction of traits with low heritability will likely require using training sets of greater size, and prediction accuracy would not be expected to reach values as high as those found in this study for highly heritable traits.As discussed above, the two sizes (20% and 10%) of the Mexican and Iranian landrace core sets gave different prediction accuracies, with the 20% core set being slightly better than the 10% core set. These results confirm the hypothesis that phenotyping and genotyping a small number of accessions and genotyping the entire population can give relatively high prediction accuracies, thereby prompting the application of GS to predict the trait value gene bank accessions for traits with relatively high heritability. The results of this study indicate that one could phenotypically evaluate only 20% of germplasm bank accessions using the reference core set of accessions for GS model training and then the values of the rest of the collection. The 20% core sets of the Mexican landraces were still large . It is possible that the GS model built on the entire phenotypically and genetically diverse core set will not predict the value of the progeny from a more restricted set of selected parents. The GS model may need to be retrained with this subset of parents.vs. the prediction core set (For genomic selection (GS), a good training set of individuals to be genotyped and phenotyped is crucial for predicting the value of candidates in the testing set. Therefore, designing a core training population that both maximizes genetic diversity and increases the accuracy of GS models is a key issue as phenotyping becomes more expensive. The method used in this study to form the diversity core was developed by core set . Better In terms of diversity measured through the Shannon Diversity Index, the expected heterozygosity  and the number of effective alleles , the two types of cores gave very similar values. For example, for the Mexican collection, the expected heterozygosity was around 0.23 for the diversity cores of both sizes, and 0.22 for the prediction cores of both sizes; the number of effective alleles for both core types of collections was around 1.4, and the Shannon Diversity Index was 0.48 for the diversity core and 0.49 for the prediction core. For the Iranian collections, the expected heterozygosity was around 0.19 for both types of core sets, and the number of effective alleles was 1.30, with a Shannon Diversity Index of around 0.5. These results indicated that forming diversity and/or prediction cores gave stability to the values of the genetic distances among accessions, and thus more diverse alleles were collected in both types of cores.In general, when accounting for population structure, genomic prediction accuracy decreases. In animal and human populations, fitting the first few initial principal components from analysis of marker data to adjust phenotypic values is a common practice for accounting for population structure. In general, population structure is important when assessing genomic prediction in stratified populations , while the latter is an introgression approach. The genetic values modeled within the landrace populations are only relevant within those populations, and thus would only be useful in a conversion strategy. Using an introgression strategy will require crossing the best accessions to elite materials, and developing a new model of the genetic effects that would be used to predict the value of future progeny. Further research is required to clarify several aspects of using genomic selection in prebreeding. Work is underway at CIMMYT to do multiple introgressions of a large number of landraces into elite materials with a large number of parents and progenies that have been initially genotyped and phenotyped. This will provide excellent populations to explore the use of genomic prediction for capturing the value present in gene banks.Results of this study indicate that genomic-enabled prediction of large wheat gene bank collections comprising Mexican and Iranian landraces for highly heritable traits may be a valuable tool for germplasm enhancement. Prediction accuracies based on random cross-validation partitions of traits related to wheat grain quality measured in one environment were intermediate to relatively high (from 0.4 to 0.6) for Mexican (7 traits) and Iranian (8 traits) landraces. Reference diversity and prediction core sets of sizes 10% and 20% of the total number of accessions gave good prediction accuracy, especially cores of size 20% that achieved similar prediction accuracies as those obtained by random cross-validation set (TRT20-TST80). Core sets of 10% had a substantial decrease in prediction accuracy (up to 16%) as compared with the prediction accuracy of the 20% core sets.Traits DTH and DTM were evaluated in drought and heat environments. Genomic models including genomic × environment (G × E) interaction gave substantial and consistent increases in prediction accuracy over the main effect models though the accession × environment interaction did not account for much variability. For Mexican landraces, models including G × E had the highest prediction accuracies, reaching correlations of 0.59–0.60 for both traits when 80% or 90% of the total number of accessions were predicted across environments when accounting for population structure. Iranian landraces had similar but slightly lower prediction accuracies. When not accounting for population structure, traits DTH and DTM for a large number of Mexican accessions were predicted with accuracies of 0.7–0.75.This study used extensive wheat landrace data stored in the CIMMYT gene bank, and has shown promising results in terms of prediction accuracies of highly heritable traits that should stimulate further research on utilizing gene bank accessions with high-density markers for future application of genomic prediction in germplasm enhancement programs.


Whole-genome sequence data is expected to capture genetic variation more completely than common genotyping panels. Our objective was to compare the proportion of variance explained and the accuracy of genomic prediction by using imputed sequence data or preselected SNPs from a genome-wide association study (GWAS) with imputed whole-genome sequence data.Phenotypes were available for 5503 Holstein–Friesian bulls. Genotypes were imputed up to whole-genome sequence  by using run 4 of the 1000 bull genomes project. The program GCTA was used to perform GWAS for protein yield (PY), somatic cell score (SCS) and interval from first to last insemination (IFL). From the GWAS, subsets of variants were selected and genomic relationship matrices (GRM) were used to estimate the variance explained in 2087 validation animals and to evaluate the genomic prediction ability. Finally, two GRM were fitted together in several models to evaluate the effect of selected variants that were in competition with all the other variants.The GRM based on full sequence data explained only marginally more genetic variation than that based on common SNP panels: for PY, SCS and IFL, genomic heritability improved from 0.81 to 0.83, 0.83 to 0.87 and 0.69 to 0.72, respectively. Sequence data also helped to identify more variants linked to quantitative trait loci and resulted in clearer GWAS peaks across the genome. The proportion of total variance explained by the selected variants combined in a GRM was considerably smaller than that explained by all variants . When selected variants were used, accuracy of genomic predictions decreased and bias increased.Although 35 to 42 variants were detected that together explained 13 to 19% of the total variance (18 to 23% of the genetic variance) when fitted alone, there was no advantage in using dense sequence information for genomic prediction in the Holstein data used in our study. Detection and selection of variants within a single breed are difficult due to long-range linkage disequilibrium. Stringent selection of variants resulted in more biased genomic predictions, although this might be due to the training population being the same dataset from which the selected variants were identified.The online version of this article (doi:10.1186/s12711-016-0274-1) contains supplementary material, which is available to authorized users. Genomic selection is increasingly applied in breeding programs for livestock species, e.g. , 2, and Several reasons may explain why the accuracy of genomic predictions does not increase when sequence data is used: (1) if the number of training individuals is small, the effects of QTL may be estimated with too large errors and thus, little advantage is gained by using sequence data ; (2) if These previous studies raise several questions i.e. how much of the total genetic variation is tagged by using different sets of variants, from commercial SNP chips up to whole-genome sequence data and to variants selected from a genome-wide association study (GWAS) using (imputed) sequence data, and how is accuracy of genomic prediction affected. Our objective was to compare the proportion of variance explained and the accuracy of genomic prediction based on imputed sequence data, lower density SNP panels, and preselected variants from a GWAS based on imputed whole-genome sequence.De-regressed proofs (DRP) were available for somatic cell score (SCS), interval between first and last insemination (IFL), and protein yield (PY) for 5503 Holstein–Friesian bulls provided by CRV . DRP were calculated according to :\documenf a bull  and was mentclasspt{minimaELdam)/4 . As the Following van Binsbergen et al. , the bulIn total, 551 of the bulls in this study were genotyped with the Illumina BovineHD BeadChip  and the other 4952 bulls were genotyped with a 50k SNP panel and imputed to BovineHD . Imputation from the 50k panel to the BovineHD SNP panel was performed with BEAGLE 3.3.0 , 16, usi1 is a vector of ones, GRM is the genomic relationship matrix calculated from the variants present on the BovineHD chip, and GRM were calculated following [For the three traits analyzed, the bulls in the discovery set were used to perform a GWAS in which each variant was fitted separately and a genomic relationship matrix (GRM) based on the BovineHD SNPs was constructed to account for population structure. The mixed linear model based association analysis (MLMA) in the package GCTA  was usedollowing , 23 as:\10(p) less than 1 for computational ease.Manhattan plots were created using the R package qqman  excludinp value based” using arbitrary cut-off levels of 3 and 5 for the −log10(p). A disadvantage is that selecting variants purely on a −log10(p) threshold results in many variants in one region being selected for genomic prediction, due to LD between variants and to running regression on each variant separately. Therefore, we also carried out a conditional and joint GWAS (COJO) using the results from the single-variant GWAS model [10(p) less than 3 were a priori removed and variants that were in LD (defined as r2) with the variants already in the model with an r2 higher than 0.8 were not added in the forward selection step. Then, variants were selected based on a conditional and joint significance level −log10(p) greater than 3 (COJO3) or 5 (COJO5). For the third and fourth COJO analyses, all variants were considered a priori, but it was assumed that variants that were more than 100 Mb apart were in complete linkage equilibrium. Thus, estimated effects were not adjusted for loci that were more than 100 Mb apart since they are assumed to be independent, and the LD threshold was only applied for loci that were less than 100 Mb apart. Also, variants that were in LD with the variants already in the model with an r2 higher than 0.5 were not added. Then, variants were selected based on a conditional and joint significance level −log10(p) higher than 5 (COJO5LD) or the 100 variants with the largest effect were selected (COJO#100).From the full set of 13,789,029 imputed sequence variants (ISQ), SNPs that were present on the two commonly used SNP panels (50k or Bovine HD) were selected as subsets. Then, the GWAS results for the discovery population were used to select 10 subsets of variants within ISQ, HD and 50k variants, which totalled 33 sets of variants. Selection was “AS model . In the 2) [GRMc. For the “p value based variant selection”   as relev  as releollowing  as descrmentclass2pt{minimvariants , and theFinally, each http://pngu.mgh.harvard.edu/purcell/plink/; [The second method to evaluate the value of the selected variants consisted in calculating genomic predictions for the validation animals (GEBV) and correlating these with the phenotypes of the same animals. Since GCTA excluded by default the animals without phenotypes from the /plink/; ), the BLAccuracy of genomic prediction was calculated for the validation animals as the correlation between the DRP and GEBV for the different traits, assuming that the DRP was based on very many progeny. Furthermore, the regression coefficient of the DRP on the GEBV was calculated to evaluate the bias of predictions.diacylglycerol O-acyltransferase 1 (DGAT1) gene, and each of the three traits, were associated with several additional variants (−log10(p) > 5) when using the imputed sequence data. The total number of variants with a −log10(p) value higher than 3 or 5 differed only slightly between traits when using the sequence data (Table 10(p) > 3 and −log10(p) > 5 respectively). If only the SNPs on the 50k and HD panels were considered, more SNPs with a −log10(p) higher than 3 and 5 were found for PY than for SCS and IFL.The GWAS results for PY, SCS and IFL are in the Manhattan plots in Figs. 10(p) > 3), 195 and 264 variants were retained for IFL and SCS, respectively, while for PY only 90 variants were selected (Table 10(p) > 5), between 35 and 42 variants were selected and their number for each trait varied little. Many of the remaining variants within the 2-Mb window on either side of the selected variants were in significant LD with the selected variants . When performing the COJO3 analysis (preselection of variants with a −log2) of 0.83, 0.87 and 0.72 for PY, SCS and IFL for the validation animals , which indicates that little additional genetic variance is picked up by the sequence data.The 10(p) value resulted in fewer variants being used for the 10(p) > 5 from the full sequence data for PY and SCS; in this case, h2 increased with fewer variants compared to when the variants were selected with a −log10(p) > 3. However, estimated phenotypic variances were highly inflated for this scenario  higher than 3, the estimated phenotypic variances were inflated but to a much lesser extent. This inflation of the h2 was probably due to the properties of the 10(p) > 5 variants with the 10(p) > 5 2 were not inflated, for example the 2 that ranged from 0.17 to 0.23 in the validation population. Still, it was clear that even the variants selected from GWAS with (imputed) sequence data could not compete in terms of variance explained with a Selection of variants based on the −log2 was lost when the selected variants were discarded from the 2 that was less than 0.01 compared to fitting the full sequence data. With the COJO scenario, the drop in h2 compared to fitting the full sequence data was less than 0.04 (results not shown).Another way of testing the importance of selecting variants was to investigate how much h2 was close to the total genetic variance that was explained by the full sequence information , except in the analysis with the HD 2 for PY and IFL. Separately, the HD 2. In comparison with the p value-based selection, the COJO scenarios resulted in lower h2 since many variants were removed from the 2 was smaller than the total genetic variance that was explained by the full sequence information, but the selected variants from the GWAS accounted for 6 to 18% of the total genetic variance, when they were estimated conditional on the To judge the relative importance of the el Table . For PY,Using ISQ variants or any of the two SNP panels resulted in the same prediction accuracy for PY (0.68), SCS (0.70 to 0.71) and IFL (0.60) Table . CompareGRM separately from the 2, and is in contrast to that of the scenarios fitting −log10(p) > 3 separately, which resulted in lower accuracy and more bias.When both 2 that ranged from 0.13 to 0.19 when fitted alone, and from 0.05 to 0.08 when the The objective of this study was to identify the usefulness of imputed sequence data, in particular regarding the proportion of variance explained by different sets of variants for three traits PY, SCS and IFL, and for the accuracy of genomic prediction. Full imputed sequence data, lower density SNP panels, and preselected variants from GWAS that used imputed whole-genome sequence were considered. Using the  Project . This stThe fact that sequence data did not improve genomic prediction was previously reported using the same data , 28. In overcome , 29, andovercome . Our res MacLeod . Hence,  MacLeod  or when  MacLeod .Our study showed that the bias in genomic prediction became stronger when variants were strongly preselected. Regressions of DRP on GEBV were expected to be 1, but we found that the stronger the selection of variants was, the stronger the bias was when comparing the predictions in the validation animals. When all ISQ variants or the SNPs on the common panels were fitted in a single a et al.  showed ta et al. . Bronduma et al.  reportedGEBV could be biased because reported effects of SNPs on a trait tend to be larger in magnitude than the true effects of these SNPs. This phenomenon is well-known, as discussed by Goddard et al. ; it is kAnother reason for the observed bias is that, rather than the estimates being biased, the bias comes from the overlap between discovery and validation data due to relationships between discovery and validation animals. Overlaps between the validation and discovery data cause bias due to the prediction error covariance between the phenotypes and predictions . Initialery data . Hence, very set . The conWhich of these underlying reasons is the major cause of the bias observed in our study is not clear. Also, to what extent the bias is controlled by fitting additional variants is not completely clear yet. Given the importance of validation results in practical breeding , more thought should be given on the validation of genomic breeding values within populations. For example, what are the effects of removing the sires of the validation population on the bias and accuracy when selected SNPs are used, or using separate populations for discovery, training and validation.10(p) values of the GWAS results. When a high −log10(p) threshold was maintained, variants were selected from a few chromosomal regions, but the resulting 10(p) value (to reduce the number of variants) but calculated the conditional and joint significance considering LD between all the variants across the whole genome. This variant selection was hindered by collinearity in the data. In spite of excluding from the model variants that were in LD with a r2 higher than 0.8, many variants were selected with large effects and −log10(p) values, calculated conditional on the joint effects already included in the model. Sometimes the same variants had no effect or effects close to zero −log10(p) in the first single variant analyses. Also, when comparing the LD between the variants, the COJO analysis selected few variants that were in moderate to high LD with each other and had opposite effects. Since the estimates were unrealistic, alternative approaches were tested by controlling the LD between variants more stringently. Fewer variants were selected with COJO5LD and only using these in the 2 and prediction accuracy, mainly because fewer variants were used. However, combining the COJO5LD together with the 2 and higher prediction accuracy and less bias for all three traits, than any of the other conditional and joint analyses. Thus, when selecting variants, the way that the variants are selected and LD is accounted for is important.To benefit from the use of sequence data, it needs to be combined with a careful variant selection to pinpoint the QTN , 39. DetIdeally, Bayesian variable selection methods should be able to separate out the causal variants when all fitted together. However, using the same dataset as in our study, van Binsbergen et al.  did not Brondum et al.  performeWhen only the Holstein breed is considered for the discovery of variants and prediction, there is little advantage in using dense sequence data for genomic prediction, although 35 to 42 variants were detected that explained 13 to 19% of the total variation when fitted alone, and 5 to 10% of the variance when they were in competition with other variants. With the within-breed approach, detection of variants and their selection were difficult due to LD. Selection of variants gave more biased genomic predictions. It is unclear if predictions were more biased due to estimating the effects of selected SNPs in the same training population as that from which SNPs were selected, or if the validation was biased due to common family structure or to the use of common data in the national analyses between the training and validation sets.


With increasing application of pooled‐sequencing approaches to population genomics robust methods are needed to accurately quantify allele frequency differences between populations. Identifying consistent differences across stratified populations can allow us to detect genomic regions under selection and that differ between populations with different histories or attributes. Current popular statistical tests are easily implemented in widely available software tools which make them simple for researchers to apply. However, there are potential problems with the way such tests are used, which means that underlying assumptions about the data are frequently violated.GLMs] with quasibinomial error structure) with attractive properties for the analysis of allele frequency differences and re‐analyse a published dataset.These problems are highlighted by simulation of simple but realistic population genetic models of neutral evolution and the performance of different tests are assessed. We present alternative tests (including Generalised Linear Models [The simulations show that common statistical tests for consistent allele frequency differences perform poorly, with high false positive rates. Applying tests that do not confound heterogeneity and main effects significantly improves inference. Variation in sequencing coverage likely produces many false positives and re‐scaling allele frequencies to counts out of a common value or an effective sample size reduces this effect.GLMs perform better and also have the attractive feature of allowing correction for multiple testing by standard procedures and are easily extended to other designs.Many researchers are interested in identifying allele frequencies that vary consistently across replicates to identify loci underlying phenotypic responses to selection or natural variation in phenotypes. Popular methods that have been suggested for this task perform poorly in simulations. Overall, quasibinomial  This consistency is important because it provides a criterion to identify alleles that underlie the same trait in many populations and to distinguish consistent responses to selection from idiosyncratic responses or effects of drift in experimental evolution studies. A hypothetical example is where three replicate lines of a large mass selection treatment are set up from three separate but identical base line populations and allowed to evolve for several generations. Pooled whole genome sequencing  which make them routine to apply. However, here we find serious consequences of the misapplication of these tests that arise from two main sources. First, heterogeneity in allele frequency differences (e.g. arising from genetic drift) is often confused for significant main effects. Second, very little attention has been paid to pseudoreplication of allele counts that is inherent in pool‐seq experimental designs (where single chromosomes are “counted” multiple times). We show that these violations of statistical assumptions produce high false discovery rates (FDRs). These problems are highlighted by simulation and we present alternative tests for the analysis which improve inference.Many of the statistical tests applicable to this kind of scenario are implemented in popular population genomic software tools  in different treatment lines or strata (Y) while k represents the number of biological replicates   which measures the magnitude of the association between the factors X and Y at level k.The most widely used statistical method to compare allele frequencies is the Cochran‐Mantel‐Haenszel test Cochran, , an exteAgresti, . In the Agresti, . A 2 × 2IfX and Y is the same at each level (k) of Z, and we are justified in describing the association with a single common odds ratio which can be tested for differences to 1  studies. Several such studies have considered data from a base population and three replicate treatment lines of Drosophila C virus (DCV). Four replicates of three regimes were compared; ancestral, sham‐control and infected, where flies were either not pricked, pricked with a sterile needle, or pricked with DCV respectively . This approach estimates the effects of a trait of interest, population of origin as well as their interaction on the allele or read count. This is similar to approaches that identify differential expression in RNA‐sequencing (RNA‐seq) experiments  from the ancestral population. We assume that the ancestral population is not under selection and is at mutation‐drift equilibrium. Thus, the ‘A’ allele frequency in the ancestral population (pA) is drawn from a beta distribution B where:u is the forward mutation rate and v is the backward mutation rate between the two states of a biallelic SNP respectively and Ne is the effective population size  is generated as a sample from a truncated normal distribution bounded between 0 and 1 , where FST represents the amount of neutral divergence from the ancestral population (FST) after t generations . 1,000,000  independent SNP datasets are generated across Balding,  with meaBalding, . No SNPs2N = n alleles is drawn from each treatment line using the binomial distribution B so as to obtain the count (x) of the ‘A’ alleles in the sample. The count of ‘a’ alleles in the pool is then n−x and the frequency of ‘A’ in the pool (fApool) is x/n. Finally, the counts among all the sequenced reads is given by another round of binomial sampling using fApool and total coverage (CT) by B. CT is either sampled or fixed and the allele counts (CA and Ca) reflect two rounds of binomial sampling (sampling from the treatment line and sampling from among the chromosomes with replacement). Data are thus generated by progressively filling the cells of contingency tables . Each partial table represents a separate replicated pair of experimental evolution lines.Because the entire population is rarely analysed in experiments a sample allele frequency at each locus of size CT across genomic regions, and treatment lines due to differences in sequencing efficiency or random variation in coverage. Because a pool contains a fixed number of chromosomes (n) and CT can often exceed n, CA and Ca are not independent draws, some chromosomes will be sampled more than once. This should be considered pseudoreplication   is 2N and neff is given by:In pool‐seq data allele counts are commonly derived from raw read counts at each locus. This can lead to substantial variation in i et al.  and Fedei et al. .u and v) values between 2 × 10−9 and 1 × 10−8 are common in e.g. Heliconius melpomene . Neutral differentiation (FST) will also depend on the population size. Here, we simulate data using values of 0.1, 0.2, or 0.3 for F,ST which is probably conservative. Only results for FST = 0.2 are given below, results for FST = 0.1 and 0.3 are shown in the supplementary material. We assume a pool size (N) of 100 throughout which is on the same order of magnitude as other experimental evolution studies  of different statistical tests. Under a null hypothesis a well‐behaved statistical test should produce a uniform distribution of  Storey, . Thus, fnd Rohlf , 1981, ap‐value distribution.While the main aim of this study is to evaluate the FPRs of these statistical tests, the ability of tests to identify true positives is also of interest. Thus, 1% of the 1 million SNPs  are designated ‘true positives’. For each true positive the SNP frequencies are simulated as above with the exception that one treatment is given a consistent allele frequency increase in 0.2 on top of whatever change is generated by drift. The ‘True Positive Rate’ (TPR) can then be roughly assessed by estimating the proportion of all true positives recovered among the bottom 1% SNPs of the 2.2stats’ package. This same function is used in the popular software package popoolation2  and an overall significant effect of treatment line (L) on allele frequency, this is tested by model structure (1). Model structure (2) simply tests whether there is an overall effect of treatment. Inconsistent allele frequency differences should increase variance in one treatment and give non‐significant treatment effects. Under model structure (1) p‐values for the treatment and interaction effects are obtained from likelihood ratio tests. For model structure (2) p‐values are from t‐tests. Counts of zero are tolerated by the GLM but can lead to other problems due to fitted values from the link function being undefined. To counter this, a common procedure is to add a count of one to each allele count if any zero counts are encountered, which we adopt here. Quasibinomial GLMs are also fitted with the glm function . Only the model structure (2), see above, is tested because there are not enough residual degrees of freedom to test for interaction effects. Interaction effects are estimated for binomial GLMs because dispersion is assumed to be 1. However, these estimates should be treated with a degree of caution. For quasibinomial GLMs, e is a quasibinomially distributed error term and p‐values for the treatment effects are obtained from t‐tests. Finally, a general Linear Model (LM) is implemented with model structure (2) in the function lm. In the LM, e is the Gaussian error term. p‐values for the treatment effects are obtained by t tests.GLMs are run in the standard R glm function, from the ‘2.4G‐tests are performed as described in . These data are re‐analysed using quasibinomial GLMs as above. The original data analysis is described in Orozco‐terWengel et al.  and 37th generation (F37) from the experiment are compared across three replicated experimental evolution lines in order to identify consistent allele frequency differences across the three replicates.Data from the E&R study on adaptation to novel temperature environments in l et al. ,2012b, al et al.  and Iranl et al. . Here, wl et al. . The minl et al. ,2012b (m33.1SD) of these differences are shown in Figures SD can be viewed as a measure of how consistent the difference between the two treatment groups is. The SD is inexact since its calculation requires a pairing of treatment lines while some statistical tests do not assume a pairing and in many experimental designs no meaningful pairing exists. There is no systematic relationship between the mean allele frequency difference and the SD of allele frequency differences indicating that these are varying freely in the simulations.The distributions of the mean allele frequency difference between the lines and the standard deviation   Figure . In term Figures  and S7.3.3st percentile of the p‐value distributions (hereafter the ‘top 1% of SNPs’). In general the CMH‐test seems to perform quite well recovering between ~15% and 29% of true positives. However, quasibinomial GLMs and LMs perform better, recovering ~30% of true positives among the top 1% of SNPs . A random sample of 100 of the SNPs that are significant after Bonferroni correction suggest that these high scoring SNPs still show patterns that researchers would want to identify, i.e. they show a consistent difference between the two time points across replicates  are far more liberal and produce more ‘significant’ SNPs  increases the TPR. In addition, as in other simulation studies , varies across the tests and simulations. Quasibinomial GLMs and LMs perform best, recovering more true positives than other statistical tests. As expected, keeping the difference applied at true positives the same while reducing the neutral differentiation  or to the effective sample size Re‐analysis of the Orozco‐terWengel et al. ,2012b dap‐values. The effective sample size procedure seems useful and is well grounded in theory  also plays a role. It is important to appreciate that several studies have used the CMH‐test and successfully uncovered biologically meaningful loci confirmed by additional functional analyses  Throughout this study we have followed the convention that the more important loci to identify are those which diverge consistently across replicate treatment lines. It is commonly argued that such loci are those most likely to represent responses to divergent selection, because inconsistent divergence may be due to drift. However, it is probably worth noting that evolutionary responses can often be opportunistic. Different SNPs segregating within genes or regulatory regions may provide alternate responses to similar selection pressures or some forms of selection  may be particularly likely to cause inconsistent responses. Hence, not all loci showing inconsistent responses in real datasets will be false positives.R.A.W.W., M.B.M., O.E.G. and M.G.R. conceived the ideas and designed the simulation methodology; R.A.W.W. performed the analyses and simulations. R.A.W.W. led the writing of the manuscript. All authors contributed critically to the drafts and gave final approval for publication. Click here for additional data file.


Although the transmission disequilibrium test appears confounded by imputation of single nucleotide polymorphisms, the pedigree disequilibrium test and family-based association test seem to offer more robust alternatives when searching for transmission ratio distortion loci in whole genome sequence data from extended families.A statistical departure from Mendel’s law of segregation is known as transmission ratio distortion. Although well documented in many other organisms, the extent of transmission ratio distortion and its influence in the human genome remains incomplete. Using Genetic Analysis Workshop 19 whole genome sequence data from 20 large Mexican American pedigrees, our goal was to identify potentially distorted regions in the genome using family-based association methods such as the transmission disequilibrium test, the pedigree disequilibrium test, and the family-based association test. Preliminary results showed an unusually high number of transmission ratio distortion signals identified by the transmission disequilibrium test, but this phenomenon could not be replicated by the pedigree disequilibrium test or family-based association test. Applying these tests to different subsets of the data, we found the transmission disequilibrium test to be very sensitive to imputed genotypes. Regression analysis of transmission ratio distortion test  Transmission ratio distortion (TRD) refers to a significant departure from the expected Mendelian transmission of alleles from parents to offspring, which is why it can only be observed in family-based studies. Meiotic drive, gametic competition, embryo lethality, germline selection and imprinting errors are among the known biological processes that can lead to skewed transmission probabilities . Althougp values for TRD <10−10, compared to none less than this threshold when limiting the analysis to sequenced subjects only. To this end, we decided to explore possible reasons for this unexpected inflation by comparing other family-based methods of association including the pedigree disequilibrium test (PDT) 4] and FBp values was being explained by deviations from HWE, as measured by the p values from tests of HWE, while controlling also for the MAF at each SNP and the quality control measures extracted from the VCF files. Poor quality genotypes often display evidence of Hardy-Weinberg (HW) disequilibrium. For each test, we ran the following multiple linear regression model in the sequenced subset:all and nuclear subsets :The objective of the regression analysis was to determine if any of the inflation observed in the FBAT p values. We used the following quality measures from the VCF files (variable ID in parentheses): number of samples with fully called data (NS), strand bias Pearson’s correlation (STR), strand bias z-score (STZ), cycle bias z-score (CBZ), cycle bias Pearson’s correlation (CBR), base-quality inflation z-score (IOZ), ratio of base-quality inflation (IOR), alternate allele quality z-score (AOZ), alternate allele inflation score (AOI), and fraction of bases with map quality less than 10 (MQ10), less than 20 (MQ20), and less than 30 (MQ30). We first ran these models on all markers with p < 1, . Then we restricted the regressions to markers with p < 10−3. Calculations were conducted in R.In the sequenced subset, we added quality control variables in case these parameters were also strongly associated with distribution of p value significance across all methods  irrespective of the number of SNPs, method, and subset type considered. In the analysis of the sequenced subjects only, none of the quality control variables showed significant associations with the - log10 p values, probably because of good preprocessing of the sequence data.Table p values, indicating that these markers are not likely to be excluded on the basis of poor performance. Our results emphasize the caution that must be taken when using the TDT in the presence of imputed data. A more unexpected result was that the PDT and FBAT were not very sensitive to the presence of imputed data. One possible reason is that there are many informative nuclear family marker configurations that can contribute to the FBAT statistic [Although it is known that small levels of genotyping error can inflate TDT results , 6, 11, tatistic  so it doBecause TRD was estimated by ignoring the true phenotype, we also performed these tests using our combined hypertension phenotype to see if different results would have been obtained. To do so, we repeated the analysis examining only transmissions to hypertensive individuals. The effect of the phenotype definition, that is, everyone affected versus hypertension, did not yield notable differences to the previous analyses for each method and subset type (results not shown).all subset to include only the individuals with these “best”-imputed genotypes, along with the sequenced subjects. This analysis showed much less inflation of TDT significance. Of note, we also observed significantly higher apparent dosage accuracy on chromosome 21. Specifically, on chromosome 21, dosage genotypes were with 1 % of an integer value for 119 individuals, whereas on other chromosomes, estimated dosages were within 1 % of integer values for only 12 to 37 individuals. Our findings therefore suggest that the TDT might be a useful measure of imputation quality for family data. For example, if the TDT reveals similar patterns to those in Fig. To further explore the effect of imputation, we examined the dosage files provided by GAW19 and extracted the “best” imputed individuals, that is, individuals whose genotype dosages were within 1, 5, 10 or 20 % of integer values 0, 1 or 2 Fig. . We thenFuture work on this topic could focus on better understanding why imputation affects some methods more than others. It would be useful to run the same analysis on reimputed data that takes the pedigree structure into account during the prephasing step. A GAW19 contribution by Saad et al. showed that the prephasing algorithm has the greatest influence on imputation accuracy (for varying MAF), and that the best performing algorithm was one that accounted for family structure. We tried running TRD analysis on their best imputed data, though it did not yield any meaningful results as there were not enough informative families in that sample set.Although it would be of interest to investigate whether any of our TRD signals are near any previously reported genes that are known to display TRD, it might be better to perform such analysis in data that are not imputed.