Various statistical association tests have been developed to analyze NGS data; however, most focus on identifying the marginal effects of a set of genetic variants on the disease. Only a few association tests for NGS data analysis have considered the interaction effects between genes. We developed three powerful gene-based gene-gene interaction tests for testing both the main effects and the interaction effects of common, low-frequency, and common with low-frequency variant pairs between two genes (the IGOF tests) in case-control studies using NGS data. We performed a comprehensive simulation study to verify that the proposed tests had appropriate type I error rates and significantly higher power than did other interaction tests for analyzing NGS data. The tests were applied to a whole-exome sequencing dataset for autism spectrum disorder (ASD) and the significant results were evaluated in another independent ASD cohort. The IGOF tests were implemented in C++ and are available at  Next-generation sequencing (NGS) has become a popular technology used in genetic studies to identify rare variants  as well as common variants associated with complex diseases. Many statistical association tests have been developed for analyzing NGS data , a variance component test via different kernels, can be used to test for gene-gene interactions using a 2wayIX kernel, which considers two-way interactions between variants in a region  in approximately 18,000 genes, resulting in an average of approximately 100 exonic variants per gene. Therefore, there would be 10,000 pairwise interactions when testing two genes with 100 variants, but there may be only a small portion of variant pairs with interaction effects on the disease. When considering combining signals from multiple test statistics, the threshold algorithm, which selects a subset of promising signals for further testing, can be a powerful approach to aggregating signals. For example, the truncated product method, which combines p-values less than a certain threshold, is more powerful than are methods combining all p-values, such as Fisher's test  was then calculated based on the IGOFcommon and IGOFLF test p-values. We performed a comprehensive simulation study to evaluate the type I error rates for the proposed tests and to compare the power of the IGOFcombined test with that of SKAT, FGLM, and KBAC. Finally, the tests were applied to the whole-exome sequencing dataset of an autism spectrum disorder (ASD) association study, and the ten most significant gene pairs were evaluated in another replication cohort.We developed gene-based tests accommodating variants with all allele frequency spectrums to identify main effects and gene-gene interaction effects for case-control studies based on threshold algorithms. The test statistic for a variant pair was based on Pearson's goodness-of-fit (GOF) statistic, which compares the difference between the observed and expected genotype distributions at two variants in cases and controls. The GOF statistic follows a chi-square distribution when all the expected values in the contingency table are greater than 5 Yates, . TherefoN cases and M controls. For two variants where one variant has alleles A and a and the other has alleles B and b, there are 9 categories of genotypes . Assume that ni and mi are the numbers of cases and controls in category i. Pearson's GOF statistic is calculated as:C2 asymptotically follows a chi-square distribution with 8 degrees of freedom. The null hypothesis of the test is that the two variants are not associated with the disease, and a variant pair with main effects and/or interaction effects could result in the rejection of the null. Note that the calculation of fi does not assume linkage equilibrium between the two variants, and therefore the statistic could be used to test for interaction of variants in linkage disequilibrium (LD).We first review Pearson's GOF statistic, which is the basis of the IGOF test statistics. Assume we have E(ni) and E(mi) for i = 1,2,…,9 greater than or equal to 5. A modification of the truncated p-value product method is applied to the IGOFcommon test. For a pair of genes, assume there are n common variant pairs. We calculate the C2 statistics and their p-values for the n pairs. Let Ω be a set of common variant pairs with p-values less than τ. The p-value threshold τ was set as 0.05 in both Zaykin et al.  of a standard normal random variable. Note that Icommon is not the product of the truncated p-values, as calculated by Zaykin et al.  or E(mi) for i = 1,2,…,9 less than 5. A threshold algorithm is also applied to the calculation of the statistic for LF variant pairs. However, the selection of LF variant pairs for testing is not based on the p-value of the C2 statistic because the chi-square property may not hold for C2 calculated based on LF variant pairs. Instead, the selection criteria are based on including variant pairs with possible risk interaction effects. To be more specific, assume there are N cases. For an LF variant pair where one variant has major and minor alleles of A and a, respectively, and the other variant has major and minor alleles of B and b, respectively, a 3 × 3 table as shown in Table N cases can be constructed, where N alleles in the variant pair, and a 2 × 2 table in Table NAB is the number of alleles A and B in individuals carrying both alleles A and B. The construction procedures for Tables M controls. Then, MAB, MAb, MaB, and Mab are calculated for the allele counts of the variant pair in controls. A variant pair is included in the IGOF test if Nab, the minor allele count of the two variants /(b+1), where R is the number of permuted statistics > the observed statistic in the b permutations. Assume that the p-values for Icommon and ILF are pcommon and pLF, respectively. Note that if Ω (or Ψ) is an empty set, no permutations will be performed and the IGOFcommon (or IGOFLF) p-value will be 1. Similar to the extended Simes' test for combining p-values into a global test p-value  is calculated as:The null hypothesis for each of the IGOF tests is that none of the variant pairs between the two tested genes are associated with the disease. Any variants with main effects or variant pairs with interaction effects could result in the rejection of the null. Hence, the IGOF tests can detect both main effects and interaction effects between two genes.C2 is fast for a pair of variants, calculating Icommon and ILF and the permuted statistics can still be very time consuming. We propose a two-stage approach to improve the performance of the IGOF tests. In the first stage, an approximated p-value based on a theoretical distribution is rapidly calculated. Adaptive permutations to calculate the p-value will be performed only if the approximated p-value is less than an elevated significance level mα, where m is a pre-specified constant and α is the significance level. The IGOFcommon or IGOFLFp-value will be 1 if no permutation is performed.Although the calculation of k largest from n independent normal random variables is asymptotically normal  Ci in Ψ can be correlated; (2) each Ci in Ψ is not necessarily distributed as a chi-square with di degrees of freedom; and (3) Ci is pre-selected based on the assumption of risk effects. Therefore, if pD is less than mα, another method based on moment-matching,   and four sets of gene pairs . The variants were randomly selected from the 600 variants in the two genes, and they consisted of both common and LF variants. When testing the interactions between two genes that were located close together on the same chromosome, there could be LD at variants between the two genes. To evaluate the effects of LD on the type I error rates for the IGOF tests, we tested two regions with strong and weak LD. To generate two regions with strong LD, from one 30 kb region, we selected the first 50 variants as the first region and the subsequent 50 variants as the second region. Similarly, to generate two regions with weak LD, from one 30 kb region, we selected the first 50 variants as the first region and the last 50 variants in the same 30 kb region as the second region. Furthermore, we evaluated the effects of population stratification on the type I error rates. COSI was used to generate another set of 10,000 sequences based on an African ancestry. Simulated samples based on the European and African ancestries were combined for the IGOF tests. Type I error rates were calculated based on 5,000 and 10,000 simulated replicates of samples for significance levels of 1 and 0.1%, respectively. That is, the IGOF tests were performed for each replicate and the p-values for the IGOF tests from the 5,000  replicates were used to calculate the type I error rates at the 1% (0.1%) significance level. To ensure that the standard error of the p-value estimation in the adaptive permutation algorithm was less than 20% of the significance level, the maximum number of permutations b and the cutoff value r were set at 2,500 and 36, respectively, for the significance level of 1%, and b and r were set at 25,000 and 36, respectively, for the significance level of 0.1%. The values b and r for the simulation study and real data analysis for ASD were estimated based on the R script available online (http://motsingerreiflab.jigsy.com/software) from the author of the adaptive algorithm . The penetrance function for SeqSIMLA2 was specified as:0, β12, β34,…, and βij are the same as those in Equation (7), and x12, x34,…, xij are the genotype coding of the causal variants based on a given interaction model. Following Wan et al.  and j , under the additive model, xij was the multiplication of minor allele counts in the two genotypes. Under the XOR model, xij was coded as 0 for genotypes AABB, AAbb, AaBb, aaBB, and aabb, while xij was coded as 1 for the other genotypes. Under the classical epistasis model, xij was coded as 1 for genotypes AAbb, AaBb, and aaBB, while xij was coded as 0 for the other genotypes. Furthermore, under the color model, xij was coded as 1 for genotypes AABb, AAbb, AaBB, and aaBB, and was coded as 0 for the other genotypes. We further considered pure epistasis models without main effects (referred to as Scenario 3). GAMETES  was determined by 0.3|log10(MAFi × MAFj)|, similar to the function used in Wu et al. (s in Equation (8). For the pure and strict epistasis model, it was difficult in GAMETES to generate multiple LF variant pairs with pure and strict interaction effects. Therefore, only one LF variant pair with such effects was simulated. Furthermore, six common variants with MAFs >30% were specified in GAMETES to generate a complicated 6-locus pure and strict epistasis model. A total of 24 models were simulated for the power comparison and the setting for each model can be found in Table We selected various variant pairs to evaluate the power  for Scenario 2. A variant pair with MAFs of u et al. . For a ccombined test with that of SKAT and FGLM. KBAC was included in the power comparisons only when LF variant pairs had interaction effects, as KBAC aimed to identify associations in rare variants. The power for SKAT was evaluated based on the 2wayIX kernel, which considers 2-way interactions between variants. For FGLM, the parameter λ, which is a smooth parameter for the Fourier expansion, was specified as 0.01/(number of variants in the genes × sample size) and the number of basis functions was specified as 23, as suggested in the user manual of FGLM. Power was calculated based on 1,000 batches at the significance levels of 1 and 0.1%.We compared the power of the IGOFcombined test when different values of τ  were specified. Furthermore, the parameter m was specified as 10 in the aforementioned simulation settings. The type I error rates for the IGOF tests may become conservative with smaller m due to a more stringent threshold for performing the adaptive permutations. We therefore evaluated the type I error rates and power for the IGOF tests under different values of m .The threshold parameter τ was specified as 0.05 as a default value in the aforementioned simulation settings. We then evaluated the power of the IGOFcombined test accommodates both common and LF variants, variants with all allele frequency spectrums were used in our analysis. We then evaluated the 10 most significant interaction pairs identified by the IGOFcombined test in another genome-wide association study (GWAS) dataset for autism obtained from the Autism Genome Project (AGP)  networks can significantly reduce the number of gene pairs to be tested, which alleviated the multiple testing correction burden and hence increased testing power  in the threshold algorithms. As expected, approximately 5% of common variant pairs were selected for calculating the IGOF statistic, while approximately 5–9% of LF variant pairs were selected. Table C2 statistic.Table combined and SKAT still maintained high power, but the power of FGLM and KBAC decreased significantly. For Model 4 where all the causal variants have MAFs < 5%, IGOFcombined still had high power followed by KBAC and SKAT, while FGLM resulted in negligible power. Figure combined had significantly higher power than did the other tests under Models 6-8, whereas under Model 5 (the additive model), SKAT had slightly more power than IGOFcombined. Figure combined consistently had the highest power across all models, except that for Model 13, FGLM had the highest power. Typically, under the XOR model (Models 10 and 14), IGOFcombined had significantly higher power than the other tests. Interestingly, although IGOFcombined assumed risk effects when selecting LF variant pairs, it still maintained power comparable to the other tests for Models 13–16, where different directions of effects were simulated. IGOFcombined also had the highest power when a mixture of common and LF variant pairs with interaction effects were simulated (Models 17–20), as shown in Figure Figure combined under Models 5, 18, and 23 when different values of τ, the threshold for selecting common variant pairs in the test statistic, were specified at the 0.1% significance level. Models 5, 18, and 23 were selected for the power evaluation because common variants were among the disease variants in these models. Furthermore, the power for IGOFcombined was not close to 1 when τ was 0.05 in these models, which would make it easier to observe the power changes when different values of τ were specified. As seen in the Figure, the power under each model was similar with different values of τ, suggesting that the default value of 0.05 for τ was appropriate for IGOFcombined to achieve optimal power in our simulations. Table m = 1 and m = 0.1. The type I error rates were appropriate when m = 1, but they can be conservative for IGOFLF and IGOFcombined when m = 0.1. The conservative type I error rates were not surprising because when m = 0.1, permutations were not performed in a larger proportion of simulated replicates compared to m = 1 and m = 10, and the replicates without permutations had p-values of 1. Figure combined under Models 5, 18, and 23 when different values of m were specified at the 0.1% significance level. Although the type I error rate for IGOFcombined was conservative when m = 0.1, we observed similar power under different values of m for these Models.Figure combined using the ARRA dataset are shown in Table p-value of 3.30 × 10−5, which did not pass the stringent Bonferroni threshold of 2 × 10−7. Interestingly, all 10 gene pairs had IGOFcommonp-values of 1, suggesting that no common interaction pairs had individual IGOF p < 0.05. Hence, the replication of the 10 gene pairs was evaluated by IGOFLF in the AGP dataset. One gene pair—ubiquinol-cytochrome c reductase core protein II (UQCRC2) and NDUFV2 had a significant IGOFLFp-value of 0.0492. Table p-values for testing the main effects of individual genes using SKAT. The kernel “linear.weighted,” which assigned more weights to rarer variants, was used. No marginal significance was observed in the gene pair of , suggesting that the significance of the IGOF tests resulted from the interaction effect between the two genes on ASD. Table In the ASD analysis, there were 897 cases and 844 controls and 222,211 exonic variants after QC in the ARRA dataset. In the AGP dataset, there were 814,767 SNPs in 1,300 cases and 1,358 controls after QC. After imputation, there were 203,270 exonic variants, where 168,961 variants had MAFs < 5%. The tests were distributed evenly across 5 computers, while 10 parallel threads were executed on each computer. The entire analysis was completed in 1 week. The 10 most significant gene pairs identified by IGOFcommon, IGOFLF, and IGOFcombined tests, which are powerful gene-based interaction tests for common, LF, and common combined with LF variant pairs, respectively. Based on threshold algorithms, only a subset of promising interaction pairs contributed to the final IGOF statistics. Moreover, while an adaptive permutation procedure was required to obtain p-values for the IGOF tests, a two-stage strategy was used to improve the efficiency of the tests in terms of run time. Our simulation study results suggest that the IGOF tests are valid tests and can have significantly higher power than SKAT and FGLM.In this study, we developed the IGOFLF statistic by calculating wj is the weight for variant pair j. Similar to the rare-variant association tests considering main effects, wj can be determined by a function of the MAFs of the variant pair, such as the Beta distribution density function used in SKAT via the “linear.weighted” kernel. However, when we used wj = Beta, where p and q were the MAFs of the variants, and 1 and 25 were the default parameters of the Beta distribution used in SKAT, there was little impact on the power of the IGOFLF test. Hence, weighting was not used in this study for the IGOFLF test.Several rare-variant association tests considering main effects assign more weights to rarer variants in the statistics, assuming that rarer variants have larger effect sizes  to consider p-values will not undergo the adaptive permutation procedure, a significant amount of time is still required in the adaptive permutations for tests with very small p-values . Therefore, the program may not be suitable for genome-wide analyses testing gene-gene interactions between all possible pairs of genes across the genome. As demonstrated in our ASD analysis, restricting the analysis to the PPI network significantly reduced the number of gene pairs tested, so the analysis can be finished in a reasonable time frame.The IGOF tests use a two-stage approach to improve computational efficiency. Although a large portion of tests with nonsignificant One major limitation of the IGOF test is that covariates such as age, sex, and population stratification are not considered. Our simulation study results suggest that in the presence of population stratification, the IGOF test can be conservative, which may result in low statistical power. In contrast to the IGOF test, covariates can be modeled in the regression-based tests SKAT and FGLM. Hence, if there is population stratification in the sample, regression-based methods such as SKAT and FGLM should be used. Another limitation of the IGOF test is that for data with imputed genotypes, it cannot use genotype dosages to account for imputation uncertainties. One strategy for analyzing imputed genotypes in the IGOF test is to use a “best guess” genotype for an individual by selecting the imputed genotype with the maximum posterior probability. This strategy was adopted in our ASD analysis for the imputed AGP data. However, as shown in the simulation studies by Zheng et al. , power cLFp-value  in the replication AGP dataset. Rare copy number variations in the UQCRC2 gene have been found to be associated with ASD  gene pair from the 10 most significant gene pairs showed a significant IGOFhttp://igof.sourceforge.net.In conclusion, we developed three gene-based gene-gene interaction tests for analyzing NGS data in case-control studies. These tests will be very useful for identifying interaction effects between two genes on complex diseases. The software implementing the three tests is available at R-HC developed the method. R-HC and C-YK designed and performed the simulation study, analyzed the data, and wrote the manuscript.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


We show this is due in part to misreporting replication cohort-size as a maximum number, rather than per-locus one. In 39 studies accurately reporting per-locus cohort-size for attempted replication of 707 loci in samples with similar ancestry, replication rate matched expectation . In contrast, ancestry differences between replication and discovery  cause the most highly-powered decile of loci to replicate worse than expected, due to difference in linkage disequilibrium.Genome-wide association studies (GWAS) have identified hundreds of SNPs responsible for variation in human quantitative traits. However, genome-wide-significant associations often fail to replicate across independent cohorts, in apparent inconsistency with their apparent strong effects in discovery cohorts. This limited success of replication raises pervasive questions about the utility of the GWAS field. We identify all 332 studies of quantitative traits from the NHGRI-EBI GWAS Database with attempted replication. We find that the majority of studies provide insufficient data to evaluate replication rates. The remaining papers replicate significantly worse than expected  perform both discovery and replication in samples of the same ancestry; and (2) report accurate per-variant sample sizes, replicate as expected. This study provides strong, rigorous evidence for the broad reliability of genome-wide association studies. We furthermore provide a model for more efficient selection of variants as candidates for replication, as selecting variants using cursed discovery data enriches for variants with little real evidence for trait association. The tradeoff between study power per-variant and resources, along with the strategy of testing millions of variants for association, leads to study designs where many associated variants of low effect size , scales inversely with the distance between the observed test statistic and the cutoff applied to variants brought forward to replication. The bias can be solved using any standard zero-finding algorithm  do both terms meaningfully contribute to the bias estimate.Here, ted in C ). Note tTo simplify predictions of replication efficiency, we considered an independent subset of all reported loci. As we lack direct access to the genetic samples used in these studies, we extracted a subset of the variants such that no two variants in a paper are situated within one megabase of any other. This is a very simple modification of the standard clumping protocol used in GWAS studies . To prevThe concept of internal “replication” may be interpreted differently in different reports. We consider three definitions of replication for this study, to observe different characteristics of the data:α = 0.05 [“nominal”]replication at nominal replication at replication within deciles of variants [“deciles”]We specifically only consider methods in which replication is determined from the replication study alone. The nominal and Bonferroni methods are commonly used. We use the decile method to investigate the behavior of the predicted power  to repliα threshold, the number of observed replications is distributed as Poisson binomial with success probabilities equal to each individual variants’ power to replicate (see below). This is a generalization of a Binomial distribution in which each Bernoulli trial is allowed to have a known but variable success rate. We use the implementation of this distribution in R [Given a set of variants and their predicted power to replicate at a given ion in R . We furtα considered, many variants analyzed here have power of effectively, or within machine precision, 0 or 1; with our limited sample size, the convergence properties of our dataset will be undesirable, and thus we use the exact distribution at the cost of computational efficiency. This process may be considered a fitting of the model according to which the WC-corrected discovery data correctly explain the observed replication data.We note that under certain assumptions the number of replications will asymptotically be distributed normally. However, depending on the In several instances, we evaluate the effects of filtering certain subsets of papers based on various criteria, and the extent to which this causes fit criteria to return to null expectation. As this evaluation is potentially confounded by reduced statistical power, in all cases we test whether the change in p-value is significantly different from expectation under random subsampling of variants matched on total power to replicate amongst the observed variants.v for a quantitative trait under the additive model isAssuming the discovery and replication sample of a study are drawn from the same source population with shared expected effect at each variant, the power to replicate a discovered variant α threshold of p is the probability of the variant exceeding the required test statistic from the null, but under the alternative distribution which is noncentral N is the replication sample size and f is the replication allele frequency of the variant. This f should be the actual allele frequency in the replication sample; however, studies sometimes report fhapmap from the closest reference ancestry as a means of protecting patient anonymity. Overall, the predicted number of replications across all variants in a paper is the sum of the power to replicate, as a function of predicted effect size and replication sample size and frequency, across all variants analyzed.In brief, the power to detect a signal at an These papers demonstrate the coverage of population ancestries in the field of quantitative trait genetics. We report and analyze the ancestral coverage of these studies using the simplifying summary statistic of continent of ancestry , tracing generally the ancestries of the original HapMap2 populations. We include a fourth category for African American samples, the largest admixed population nonnegligibly represented in the papers. This geographical partitioning matches the ancestry assumptions used in GWAS methods such as genotype imputation.Ancestry group counts are computed from maximum reported sample size per cohort per paper. In studies where cohorts of different continental ancestry are meta-analyzed, sample sizes are appropriately partitioned to the contributing ancestries. No adjustment is applied for papers reporting on the same cohort. For comparison to what the field’s sample sizes would be under random global sampling, global population estimates are computed , 13.We tested loci for nonrandom annotations. This test is usually conducted with access to the full set of variants tested in an individual study. As in this study design such information is masked, we restricted the analysis to papers using HapMap2 imputation in their discovery data; considered only SNPs present in HapMap2; and restricted the data further to European ancestry discovery data, which includes the majority of papers in the dataset.We annotated all variants in the CEU subset of HapMap2 using ANNOVAR . We compS1 Figα. X-axis: replication α; Y-axis: estimated SNP effect from linear regression. Two simulations with different effect estimates  are shown. The height of the vertical lines corresponds to the average bias (curse) introduced by discovery prioritization.A SNP with fixed frequency, sample size, and phenotypic variance explained was repeatedly drawn at random, and the accompanying standard normal trait was simulated. The apparent effect of SNP on trait was estimated exclusively from simulations exceeding progressively more stringent replication (TIFF)Click here for additional data file.S2 Figp < 0.00001). Generated using the software at [Includes all loci brought forward from discovery, not filtered for independence. Datapoint shapes correspond to approximate continental ancestry: CEU = European, YRI = Click here for additional data file.S3 FigSubset of data from Supp. Fig; includes only loci passing study-specific Bonferroni-corrected replication threshold. Generated using the software at .(TIFF)Click here for additional data file.S4 Figp < 2 ⋅ 10−16. Considers studies with same ancestry in both discovery and replication and correct per-locus sample sizes, as used in the main results.Plotted are receiver operating characteristics (ROC) for raw discovery and WC-corrected estimates of power to nominally replicate variants in individual studies. X-axis: false positive rate, computed as number of non-replicating variants having greater than a given power to replicate; Y-axis: true positive rate, computed as number of replicating variants having such power. AUC: 0.795 (WC-corrected) versus 0.579 (biased); DeLong’s two-tailed (TIFF)Click here for additional data file.S5 FigTop panel: predicted versus expected replication for each paper, with each paper flagged as being within 95% confidence of predicted replication rate under WC-corrected model, greater than or less than expectation. X-axis: predicted number of replications in a given paper, calculated as the sum across all loci of power to replicate based on WC-corrected discovery effect estimates. Y-axis: observed number of replications in the paper. Colors correspond to publication dates binned into six month intervals, from 2008 to 2014. Point shapes to correspondence between observed and expected rates. Replication is defined as a one-tailed replication p-value surpassing a per-paper Bonferroni threshold: (TIFF)Click here for additional data file.S6 FigEach node corresponds to one method of subsetting the data: all papers; papers conducting discovery and replication in the same continental ancestry; papers reporting correct per-locus sample sizes; or papers doing both. Probabilities in nodes correspond to two-tailed Poisson binomial fit test for prediction of nominal replication rates after Winner’s Curse correction. Probabilities along edges correspond to the chance of randomly seeing improvement of fit between connected nodes at least as large as observed due to loss of power exclusively, based on 10000 simulated subsamplings from the source node matched on total predicted power to replicate. Relative position of nodes along vertical axis corresponds to number of loci removed in the subset.(TIFF)Click here for additional data file.S7 FigEach node corresponds to one method of subsetting the data: all papers; papers conducting discovery and replication in the same continental ancestry; papers reporting correct per-locus sample sizes; or papers doing both. Probabilities in nodes correspond to two-tailed Poisson binomial fit test for prediction of Bonferroni replication rates after Winner’s Curse correction. Probabilities along edges correspond to the chance of randomly seeing improvement of fit between connected nodes at least as large as observed due to loss of power exclusively, based on 10000 simulated subsamplings from the source node matched on total predicted power to replicate. Relative position of nodes along vertical axis corresponds to number of loci removed in the subset.(TIFF)Click here for additional data file.S8 FigX-axis: left boundary of bin on replication threshold; Y-axis: percentage of replication within corresponding bin. Tracks correspond to actual data in the replication study, predicted power to replicate using discovery effect estimates, and predicted power to replicate using WC-corrected discovery effect estimates. Loci brought forward to replication in papers were pruned to contain independent signals within (but not necessarily between) individual papers. Error bars correspond to 95% confidence intervals around mean replication rates as estimated across multiple loci. Left panel: including all papers; middle panel: including only papers with accurate per-locus sample sizes; right panel: including only papers with maximal sample sizes per variant. Including only maximum sample size regardless of per-variant missingness will uniformly inflate replication rate estimates.(TIFF)Click here for additional data file.S9 FigX-axis: left boundary of bin on replication threshold; Y-axis: percentage of replication within corresponding bin. Tracks correspond to actual data in the replication study, predicted power to replicate using discovery effect estimates, and predicted power to replicate using WC-corrected discovery effect estimates. Loci brought forward to replication in papers were pruned to contain independent signals within (but not necessarily between) individual papers. Error bars correspond to 95% confidence intervals around mean replication rates as estimated across multiple loci. Left panel: including all papers; middle panel: including only papers with accurate per-locus sample sizes; right panel: including only papers with maximal sample sizes per variant. Including only maximum sample size regardless of per-variant missingness will uniformly inflate replication rate estimates.(TIFF)Click here for additional data file.S10 FigX-axis: left boundary of bin on replication threshold; Y-axis: percentage of replication within corresponding bin. Tracks correspond to actual data in the replication study, predicted power to replicate using discovery effect estimates, and predicted power to replicate using WC-corrected discovery effect estimates. Loci brought forward to replication in papers were pruned to contain independent signals within (but not necessarily between) individual papers. Error bars correspond to 95% confidence intervals around mean replication rates as estimated across multiple loci. Left panel: including all papers; middle panel: including only papers with accurate per-locus sample sizes; right panel: including only papers with maximal sample sizes per variant. Including only maximum sample size regardless of per-variant missingness will uniformly inflate replication rate estimates.(TIFF)Click here for additional data file.S11 FigX-axis: left boundary of bin on replication threshold; Y-axis: percentage of replication within corresponding bin. Tracks correspond to actual data in the replication study, predicted power to replicate using discovery effect estimates, and predicted power to replicate using WC-corrected discovery effect estimates. Loci brought forward to replication in papers were pruned to contain independent signals within (but not necessarily between) individual papers. Error bars correspond to 95% confidence intervals around mean replication rates as estimated across multiple loci. Left panel: including all papers; middle panel: including only papers with accurate per-locus sample sizes; right panel: including only papers with maximal sample sizes per variant. Including only maximum sample size regardless of per-variant missingness will uniformly inflate replication rate estimates.(TIFF)Click here for additional data file.S1 FileContains: Tables A-C, and bibliography for all papers considered from the NHGRI-EBI GWAS Catalog.(PDF)Click here for additional data file.S1 DatasetEach file corresponds to a single PMID as listed in the filename. Only includes data for 100 studies used in analysis. For full citations, see (TGZ)Click here for additional data file.S2 DatasetIncluded: the raw data from (IPYNB)Click here for additional data file.


Canine hip dysplasia (CHD) and elbow dysplasia (ED) impact the health and welfare of all dogs. The first formally organized assessment scheme to improve canine health centered on reducing the prevalence of these orthopedic disorders. Phenotypic screening of joint conformation remains the currently available strategy for breeders to make selection decisions. The present study evaluated the efficacy of employing phenotypic selection on breed improvement of hips and elbows using the Orthopedic Foundation for Animals complete database spanning the 1970–2015 time period. Sixty breeds having more than 1000 unique hip evaluations and 500 elbow evaluations  were interrogated to derive phenotypic improvement, sex and age at time of assessment effects, correlation between the two joints, heritability estimates, estimated breeding values (EBV), and effectiveness of maternal/paternal selection. The data demonstrated that there has been overall improvement in hip and elbow conformation with a reduction in EBV for disease liability, although the breeds differed in the magnitude of the response to selection. Heritabilities also differed substantially across the breeds as did the correlation of the joints; in the absence of a universal association of these differences with breed size, popularity, or participation in screening, it appears that the breeds themselves vary in genetic control. There was subtle, though again breed specific, impact of sex and older ages on CHD and ED. There was greater paternal impact on a reduction of CHD. In the absence of direct genetic tests for either of these two diseases, phenotypic selection has proven to be effective. Furthermore, the data underscore that selection schemes must be breed specific and that it is likely the genetic profiles will be unique across the breeds for these two conditions. Despite the advances achieved with phenotypic selection, incorporation of EBVs into selection schemes should accelerate advances in hip and elbow improvement. Canine hip dysplasia (CHD) is a highly prevalent inherited condition affecting dogs –3 irrespFor decades, there have been concerted efforts to characterize and reduce the incidence CHD , 13. RepElbow dysplasia (ED) is another orthopedic condition long recognized in the dog  with cerRetrospective analyses of epidemiological data have revealed a genetic basis to ED with heritability estimates of 0.10 to 0.77 and males showing a greater heritability than females –32. SimiThe risk of concomitant expression of CHD and ED has been estimated to be 1.67 with an increased correlated risk when either joint has severe dysplasia  35]. Th. Th35]. The objectives of the present study were to characterize the influence of non-compulsory selection on phenotypic radiographic assessment of hip and elbow conformation over time. Dog breeds that were most highly participatory in a voluntary United States radiographic screening process for hips and elbows were evaluated for improvement, whether maternal or paternal selection was more responsible for any observed progress, whether some breeds prove more amenable to selection than others, and whether selection against one orthopedic disorder yielded concomitant improvement in the other.Data: The complete Orthopedic Foundation for Animals (OFA) database for hip and elbow ratings on dog breeds with more than 1000 records for hips and 500 records for elbows was interrogated. Data evaluated included those publically released and those withheld from public release from 1970 through April 2015. The OFA categorizes hips as excellent, good, fair, borderline, mild dysplasia, moderate dysplasia, severe dysplasia with a single rating that encompasses both hip configurations. For the analyses, hip scores were classified as follows: 1 as excellent, 2 as good, 3 as fair, 4 as borderline, 5 as mild, 6 as moderate, and 7 as severe. Hip scores 1–3 are considered non-dysplastic. Elbow scores were classified similarly with 1 as normal, 2 as degenerative joint disease grade I, 3 as degenerative joint disease grade II, and 4 as degenerative joint disease grade III.ijknm is the underlying risk variable of the m-th trait (m = hip or elbow) on the n-th dog  of the i-th sex (i = M or F), diagnosed in the k-th year  and diagnosed at the j-th age in months . μm is a constant common to all dogs for a given joint trait and sexim is the effect of sex i on trait m. βm is a regression coefficient relating the m-th trait score to the age of the dog at diagnosis. Following that are terms for the k-th year of diagnosis effect of the m-th trait and the genetic contribution each animal brings to its own hip or elbow score . The year of evaluation was computed from the data file using date of birth and age at diagnosis. Finally, eijknm represents the remaining residual, or any unexplained variability in hip and elbow scores.Hip and elbow scores are ordered categories, suggesting a threshold model with a probit link function  as a natijknm, can be represented as a vector of breeding values for all dogs identified in the data set of a given breed. This can be represented as two vectors of equal length , ah and ae, for the traits of hips and elbows, respectively. Accordingly, we assumeA being the numerator relationship matrix [G0 containing the genetic variances for hips and elbows (gh and ge) along with the genetic covariance between hips and elbows (ghe) and the symbol ⊗ representing a direct product of the two matrices G0 and A.The animal contributions for a given trait, animalp matrix , G0 contijknm, can also be represented as two vectors eh and ee for hips and elbows, respectively, random variables assumed to be uncorrelated with the genetic effects in ah and ae. The variances for each residual term  are set to 1.0, a necessary component for the fitting of mixed categorical models. However, for those individuals with observations on both traits we can consider the parameter rhe, the covariance between residuals for the risk of hip and elbow disease. The covariance structure of residuals for this subset of animals that have a recorded hips and elbow score would then take the formR0 is a matrix of residual variances  and covariance (rhe) for hips and elbows, I is an identity matrix of order the number of dogs scored for both hips and elbow disease and, as above, ⊗ signifies a direct product.In a similar fashion, the residuals for each trait, em, sexim, βm and year of diagnosiskm) and to predict the random effects , along with estimation of the unknown genetic variances , the genetic covariance (ghe) as well as the residual covariance (rhe). In addition, we can estimate the heritability of risk for hip disease as h2h = gh / (gh + 1) and that for elbow disease as h2e = ge / (ge + 1), along with the genetic correlation between these traits as rg = ghe /√(gh ge). Finally, we can also use the predicted breeding values to estimate the impact of selection in each of these traits across the 60 breeds represented in this data set.The objective was to estimate the unknown fixed effects for each trait  were independent normal densities with null means and variances of e10, that is, a diffuse normal prior. The prior densities for the putative random effects  were multivariate normal densities with null means and covariance represented in G0  was assumed to be an inverse-Wishart density. For variance components the inverse-Wishart is governed by two parameters, V and nu in the notation of [he, was also assumed to follow an inverse-Wishart density, necessarily constrained to ensure that the covariance matrix R0 was positive definite [A Bayesian framework, a strategy of considerable power and plasticity, was used to evaluate these unknown values . The pubation of . The disation of  refers tdefinite .As implied by the use of the MCMCglmm package, estimates of the posterior density for the unknown parameters were generated through a Monte Carlo Markov chain. Each trait was evaluated in one continuous chain, each starting at disperse values for the unknown variances. Convergence was examined through trace plots making use of the R package coda . Each chIn the process of estimating heritability, we are also able to extract the additive genetic merit of all dogs in each pedigree. These estimated breeding values (EBVs) can be used to evaluate the response to selection on hip and elbow disease by participating breeders. The estimation of this genetic trend, done on several of the 60 breeds represented in the data set, is no more than the arithmetic mean of the EBVs of all dogs in each year of birth. Plotting these values, though on the unobserved probit scale, does permit an assessment of the changing genetic profile of the breed over time. Data were then fitted to a loess regression within the ggplot2 package  in the pIn a few selected instances, we also chose to deconstruct the observed selection response across the four paths of possible genetic improvement that are often used in livestock improvement. That is, to source the response to selection through the differing intensities and accuracies typically occurring in the selection of sires of sires (SS), sires of dams (SD), dams of sires (DS) and dams of dams (DD). This technique was first proposed many years ago in dairy cattle . This caRestricting the OFA data to dogs 2 years of age and older at time of evaluation and to breeds with the requisite number of hip and elbow evaluations resulted in 1,056,852 and 275,129 hip and elbow records, respectively, representing 60 different breeds and in some cases more than eleven generations. The number of records that met the criteria for analyses varied by breed  with a hThe fifteen breeds with the highest proportion with dysplastic hip and elbow ratings are presented in When evaluating a sex effect on hip ratings , in 14 bIncreasing age had a subtle but significant (p < 0.05) effect in 34 breeds with older dogs having, on average slightly higher ratings for hips . Age hadHeritabilities for both joints differed by breed  with an The correlation between the two joint ratings also varied by breed with 14 breeds having correlations > 0.9 and the Alaskan malamute and Nova Scotia duck tolling retriever having nearly perfect correlation . In contrast, 19 breeds showed no significant correlation between the two joint ratings and the two joints were significantly negatively correlated in the Doberman pinscher.An intensive examination of each of the 60 breeds represented in this study is beyond the scope of this publication. Therefore, we selected several breeds for closer inspection to examine the evidence for possible phenotypic and genetic progress in combatting joint disease. We examined several breeds of high prevalence, along with a comparison to several other breeds with a low prevalence of CHD. As presented in We assessed the trend of EBV over time to provide a sense of improvement in the hip phenotype and genetic progress. The genetic and phenotypic trend in hip ratings among the high prevalence breeds are presented in To ensure that the findings observed for hip conformation were applicable broadly, breeds having the greatest number of hips assessed  that is,The selection patterns derived from evaluating the impact of maternal or paternal selection strategies on progress in CHD are shown in Minimizing the incidence of debilitating orthopedic disorders is paramount, yet past studies that have evaluated CHD and ED heritability report highly variable estimates and nominal response to selection. Contributing factors to disparate heritability estimates include sample size, breed composition, and relatedness of the population studied. The time frame under study, if the evaluation scheme is voluntary or mandatory, and if all evaluations are available for analyses also influence the accuracy of the estimate and assessment of joint improvement. In the absence of DNA-based genetic testing, utilization of estimated breeding values based upon phenotypic radiographic assessment related to CHD and ED has been promoted. Therefore, it is important to best characterize the potential improvement that can be achieved based upon phenotypic selection. The present study was undertaken to comprehensively assess the impact of phenotypic selection on hips and elbows over several decades in multiple dog breeds.In this study we determined the impact of age and sex on the expression of CHD and ED. Similar to other studies that evaluated fewer breeds and targeted individual dogs , 46, incWhen one considers the phenotypic trend of CHD across the period of available data, all breeds appear to be making progress, which is reflected in the increasing proportion of dogs scored as “excellent”. Not surprisingly the response in the breeds with greater numbers of data points  demonstrate a more stable and consistent increase in this proportion. Breeds with a smaller population size  display a trend of improvement, but with more instability across years and even a loss of gain in excellent hip phenotypes with the Bloodhound. This instability was also reflected in the changes in average breeding value across time.As expected, the consistent trend in all five high prevalence breeds, and those breeds with high submissions, showed a decline in the average EBV, remembering that on the scale used here, higher hip scores reflect increasing severity of disease and lower EBVs indicate a trend to improved hip scores. Owing largely to the number of dogs available in each birth year, Newfoundland and Rottweiler demonstrate a steady decline in average breeding value. For breeds with low CHD prevalence, the phenotypic advancement over time was greater than in the breeds with higher prevalence. This evidence suggests that breeders appear to be using the information collected in the various health databases, including health registries managed by specific breed clubs, for making informed choices on breeding alternatives.Interestingly, the Labrador retriever showed a slower rate of decline in average CHD EBV over the time period studied than that observed in other popular breeds. Yet, at the same time, the proportion of excellent-scored hips increased substantially. We believe that what is demonstrated in this seemingly inconsistent observation is a combination of selection by breeders (i.e. selection against hip disease) and a “migration” of new animals to register with the OFA database . EBVs are computed with a reference to a base population, that is, a group of animals that do not have parents or the parental phenotype identified. Including new animals without parents or ancestors in the database will necessarily re-orient the base population, impacting the average breeding values over the course of time and altering the scale of genetic progress. Nevertheless, these new “migrants” will contribute their observed phenotypic class to the fraction of animals scored as “excellent”. Of course this combination of selection progress and the inclusion of new animals into the database occurs in all populations, not only Labrador retrievers. Nevertheless, the impact of including new animals in the database with no known ancestors in the pedigree is likely exacerbated in breeds with large populations than one would expect in small population breeds where the natural accumulation of inbreeding will reflect more relationships among the animals in the database.Taking a broad view of the estimates of heritability for CHD, one might be struck by the relatively high values for this parameter across this sample of breeds, in contrast to other published estimates of this parameter in earlier investigations. The overall range of CHD heritability estimated in the present study was 0.46 to 0.76, somewhat higher than 0.35 and 0.44 reported for Labrador retrievers , 53 and One explanation for the observed heritability estimate differences may reflect individual evaluation of each breed as opposed to a composite assessment. The overall range of ED heritability similarly reflected breed differences with a high of 0.89 to a low of 0.01. Notably, breeds with low estimates did not reflect prevalence of ED nor limited numbers of dogs assessed. A similar range of ED estimates has also been reported in the literature for multiple dog breeds , 32, 57Another likely factor contributing to relatively high heritabilities is the sample of animals considered in the analysis. Dogs with data in either of these joint disorder data sets are unlikely to be a random sample of their respective breeds. Indeed, one might reasonably suspect that this sample is more likely to reflect owners interested in the impact of inheritance on these disorders and leveraging that in their breeding decisions. Of course, the method of analysis can also be added as a possible explanation for our relatively high parameter estimates. As outlined in the section on methodology, our analysis reflects the true categorical nature of the OFA scoring system, in contrast with the work of other investigators  who evaContrasting the heritability values for ED with those for CHD, the difference in range of these values is apparent (0.01 to 0.89 for elbows and 0.46 to 0.76 for hips). The most immediate cause is likely statistical, reflecting the greater number of records available on hip disease than that available for elbow disease. Sample size, compounded by the nature of those owners/breeders who actively participate in contributing records to this database, can have a profound impact on the estimation of variability. In addition, issues of sample size are also reflected in the variability of the ED phenotype itself, where some breeds have a considerable proportion of animals scored as “normal”. Finally, it is important to note, when estimating these genetic parameters in populations spanning several generations, that the estimates also take into account the level of genetic variation in the initial base population  rather than directly reflecting the present breeding population . This inThe findings of the present study demonstrate the variability of breeds and underscore the clear need to consider the individual breed when assessing disease liability. The heritabilities were also analyzed by haplotype relationship  to detecGenetic correlation is typically interpreted as a measure of pleiotropy in the loci impacting traits as well as genes that may be in linkage disequilibrium with the pleiotropic genes. Therefore, a positive genetic correlation between hip and elbow scores would be the ideal outcome for breeders, implying that as selection pressure is placed on one trait, the other will move in the same direction. Negative correlations impose a more challenging setting for the improvement of both traits simultaneously. The magnitude and direction of a genetic correlation is a reflection not just of the loci involved but also of the allele frequencies that underlie those genotypes. The variability in correlation between CHD and ED reported in the literature and identified here indicates the uniqueness to the breeds with respect to orthopedic conditions. Correlations between the two joints range from 0 to 0.32 , 66 to aPerhaps the most notable element of the phenotypic trends in ED displayed by the data is, with the exception of the Chow chow breed, the consistent proportion of dogs scored as normal, even within the breeds having a high prevalence of ED. Nevertheless, there was a steady, if small, decline in the average breeding value for ED, appearing to begin with those dogs born near the year 2000. This slight improvement in the genetic merit towards improved elbow conformation may have been associated with the selection pressure placed on hips; this being a correlated response in elbow health mediated by the positive genetic correlation between hips and elbows in these five breeds refer to . Among bThe breeds having a high prevalence CHD all had statistically equivalent heritability estimates for CHD, yet displayed differential reductions in EBVs over time. Similarly, the low prevalence CHD breeds had variable heritability estimates ranging from 0.49 to 0.61 without demonstrating a relationship between improvements in EBVs that might be ascribed to heritability. The same lack of direct relationship of heritability estimate and EBV response was noted for ED. Although one might expect that the breeds with the largest heritabilities would exhibit the greatest response, intensity of selection, population size, use of a popular sire, and selective owner participation in a non-compulsory evaluation scheme all play a role in the EBV calculation.As with any study, there are limitations to the present analyses. The data reflect only those dogs whose breeders and owners submitted radiographs for analysis by the OFA. An ideal data set would include hip and elbow evaluations for parents and all offspring for all the breeds. The representation of the present population varied by breed; for some breeds the submission rate represented a much higher proportion of the purebred population than for others. A study by Paster et al.  suggesteGlobally, CHD is a recognized welfare concern and efforts have been in place for several decades to reduce the prevalence. Ginja et al., 2010 recommend implementing concerted EBV breeding schemes to reduce incidence; these authors also stated that individual dog phenotypic selection has been ineffective in producing change , 37, thaMany investigators have advocated the implementation of estimated breeding values (EBV) to accelerate progress in the selection process , 69–71 sThe present data indicate the effectiveness of concerted selection schemes based upon phenotypic assessment in improving joint health and would suggest emphasizing selection on sires, a practice currently employed by breeders. One may argue that sires produce more offspring and may skew the interpretation. However, importantly, the four path plot analyses that demonstrate the role of sire selection are not weighted by the number of grand-progeny a given dog has in the population. Thus, the data suggest that more rapid improvement may be achieved by emphasizing the health characteristics of the sires. One must be mindful of avoiding a popular sire effect. Popular sires are the greatest contributor to the spread of genetic disorders  yet prudThe data presented here confirms that employing phenotypic health information and selecting sires and dams from pedigrees free from dysplasia does reduce the condition. Acceptance of using health information in breeding decisions is growing. A 2004 study of Dutch Boxer breeders  indicateS1 Table(PDF)Click here for additional data file.S2 Table(PDF)Click here for additional data file.S3 Table(XLSX)Click here for additional data file.


We observed high concordance between true and imputed marker scores and the HMM produced well-calibrated genotype probabilities that correctly reflected the uncertainty of the imputed scores. Genomic estimated breeding values (GEBV) calculated from the imputed scores closely matched GEBV calculated from the true marker scores. The within-population correlation between these sets of GEBV approached 0.95 at 1x and 4x coverage when pooling two or four individuals, respectively. Our approach can reduce the genotyping cost per individual by a factor up to the number of pooled individuals in GBS applications without the need for extra sequencing coverage, thereby enabling cost-effective large scale genotyping for applications such as WGS in plant breeding.The increased usage of whole-genome selection (WGS) and other molecular evaluation methods in plant breeding relies on the ability to genotype a very large number of untested individuals in each breeding cycle. Many plant breeding programs evaluate large biparental populations of homozygous individuals derived from homozygous parent inbred lines. This structure lends itself to parent-progeny imputation, which transfers the genotype scores of the parents to progeny individuals that are genotyped for a much smaller number of loci. Here we introduce a parent-progeny imputation method that infers individual genotypes from non-barcoded pooled samples of DNA of multiple individuals using a Hidden Markov Model (HMM). We demonstrate the method for pools of simulated maize double haploids (DH) from biparental populations, genotyped using a genotyping by sequencing (GBS) approach for 3,000 loci at 0.125 With the advent of whole-genome evaluation and other molecular methods in plant breeding –3, the aGenotype imputation is a promising and well-studied approach to reduce genotyping costs . ImputatObtaining genotypes from DNA sequence data, termed genotyping by sequencing (GBS) , emergedIn contrast, methods that do not require individual sample identity achieve cost reduction by pooled genotyping, which combines DNA from several individuals and genotypes them jointly in a single assay . Pooled Parent-progeny genotype imputation from pooled samples infers the marker locus genotypes of the pooled individuals in reference to the set of their direct ancestors . This requires as input the following four pieces of informationthe complete marker genotypes of the parents at all loci of interestthe genotype of the pooled DNA sample (possibly at only a subset of the marker loci)the pedigree relationship between the pooled individuals and their parents, andthe genetic linkage map of the lociGiven this information, we calculate for each locus the posterior probabilities of the identity by decent (IBD) inheritance configurations which describe possible patterns of inheritance from parent to offspring. These probabilities are then used to infer the marker genotypes of the pooled individuals. Hereafter we will use the term ‘imputation’ to indicate the inference of genotype scores of individuals from pooled DNA samples, regardless of whether a marker genotype was observed in the pool or not.1 and P2) at four biallelic SNP markers . The DH are progeny of two biparental populations with four distinct and fully homozygous inbred lines as parents (I1 × I2 and I3 × I4). The DNA of the two DH is pooled into a single sample and genotyped. The critical task becomes inference, at each locus, of the parent of origin for each DH in the pool. We term the combination of parents of origin the inheritance pattern of a locus and denote it as, e.g., I1-I3.The following example will introduce the concept intuitively . Assume Our example incorporates the read counts of each allele of a marker as would be available if a GBS method is used for genotyping. For simplicity, the parent genotypes at each locus are recoded to represent dosage of a chosen reference allele such that ‘0’ represents a diploid individual with no doses of the reference allele , and ‘2’ represents a diploid individual with two doses of the reference allele (homozygous reference). The genotype data for the pool then becomes the sequence read counts of the reference allele relative to the total read count. We will henceforth denote pool genotypes in which only a single allele is present as “homogeneous” and those with multiple alleles as “heterogeneous”, in distinction to homozygous or heterozygous genotypes of individuals. A key factor for inference is the ability to assess whether a pool presents a homogeneous or heterogeneous allelic state at each locus. The technique of inference is therefore not limited to sequencing methods, as any genotyping approach that can detect allelic heterogeneity in the DNA pool would suffice.1, with one read of the reference allele out of three total reads. In the absence of genotyping error, the true inheritance pattern must therefore contain both marker alleles. At this locus only parent I4 carries the alternate allele and only DH P2 can inherit from this parent. Consequently, P2 must carry the alternate allele and P1 the reference allele. This inference was made possible by knowledge of the parental genotypes and of the pedigree linking parents to DH progeny. A similar reasoning can be applied to locus L4 to infer that P2 carries the reference allele. A heterogeneous genotype was also detected at locus L2. Here, however, the pedigree and genotype information are inconclusive on their own because both the reference and alternate alleles could each be traced to either DH pedigree. For example, the same observed pool genotype could have arisen from P1 and P2 inheriting respectively from either I1 and I4 or from I2 and I3. Although both scenarios are equally likely when locus L2 is considered on its own, their relative probabilities can be updated with information from linked loci. Having established the marker genotypes at loci L1 and L4, and with knowledge of the genetic distance between the loci, it can be shown that the second inheritance pattern (I2 and I3) is the more likely one because it requires a recombination within a 10cM interval instead of a 7cM interval. Thus, the most likely genotype at L2 is the reference allele for P2 and the alternate allele for P1. We are then left with locus L3, for which no read counts were observed for either allele. By combining all of the aforementioned information, it can be shown that again I2-I3 is the most likely inheritance pattern, because it does not require any additional recombination events beyond the one invoked previously. It follows that P1 most likely inherited the alternate allele and P2 the reference allele at L3. The purpose of this small example was to show how loci with multiple possible inheritance patterns or missing data can be resolved by collectively weighing information from the genetic linkage map, the marker genotypes at linked loci, and the pedigree. Such heuristic reasoning is clearly impractical for more than a few loci and facilitates only very crude inference. A more formal and powerful approach will be described next.In our example, a heterogeneous pool genotype was detected for marker LIf all parents of the pooled DH are present in the ancestor set and the pedigree fully describes all crosses carried out, then the sequence of inheritance patterns along the genomes of the pooled offspring fulfills the requirements of a Hidden Markov Model (HMM). The HMM incorporates the four pieces of information outlined above in the form of the emission and the transition matrix. The emission matrix provides the probabilities that an observed pool genotype could be produced by each possible hidden state of the ancestral inheritance pattern. The transition matrix provides the probabilities that the inheritance pattern at the previous locus can result in a particular pattern at the current locus. These probabilities are a function of both the pedigree and the genetic map. Throughout we assume that the parents of the pooled individuals are fully homozygous inbred lines.k, with an emission matrix Ek, a transition matrix Tk, and a vector of forward probabilities from the previous step (henceforth denoted as fk−1), the forward pass ism,] specifies the row of the emission matrix for the observed genotype m , ‘∘’ refers to element-wise multiplication, and ck is a normalization constant equal to bk indicates the vector of backward probabilities and ak is similarly defined as ck.The forward-backward algorithm  providesf0, which is used when k = 1, corresponds to the prior probabilities for the populations involved in the pool. For a pool of two DH from a biparental, F1 derived population 1 crosses). The initial bM+1, where M is the number of markers, for the backward pass is always The initial vector of forward probabilities k = 1 to k = M and the backward pass from k = M to k = 1. The posterior inheritance probabilities at locus k are then obtained by calculatingThe forward pass is executed from Tk for locus k for a pool of two F1 derived DH from fully homozygous parents isrk is the recombination frequency between loci k and k−1. For example, the value in row 1 column 2 of this matrix describes the probability that P1 inherited from parent I1 and P2 from parent I4 at locus L2, conditional on the two progeny inheriting from I1 and I3, respectively, at locus L1. For P1 this requires that there is no recombination between the two loci, which happens with probability (1 − rk). For P2, the transition from I3 to I4 requires a recombination event, which has probability rk. Because both events happen independent of each other, the joint probability is rk(1 − rk). The same rationale can be applied to derive transition matrices for different cross types . Progeny from advanced crosses with additional rounds of meiosis  can also be modeled appropriately. Similar to other parent-progeny approaches, the reduction of linkage between markers in an advanced cross design could lead to lower imputation accuracy unless marker density is increased.We will now use the previously introduced example in r the F1 ) or to pEk for locus k describes the probability of observing a marker genotype conditional on the inheritance pattern at that locus. The genotype data generated by most sequencing platforms is observed in the form of allele counts and can be modeled with a Beta-Binomial probability distribution. Briefly, the Beta-Binomial distribution models the probability of observing m reads of a reference allele out of n total reads, when the underlying allele frequency in the sample is uncertain. In principle, this allele frequency is determined by the genotypes of the parents involved in a particular inheritance pattern and can be calculated easily. However, technical variation in quantity and quality of the DNA that each individual contributes to a pool can distort allele frequencies and generate uncertainty [The emission matrix ertainty .m reference allele reads out of n total reads isB is the Beta function and α and β are positive parameters that reflect the uncertainty in the reference allele frequency. The average frequency is given by α/(α + β) and the smaller α + β, the more variation is expected around it. The parameters were calculated as follows:π is the expected or estimated frequency of the reference allele  for a given inheritance pattern and ν a dispersion parameter reflecting the uncertainty in the estimate. A smaller value for ν implies greater uncertainty, with ν > 0 .Under the Beta-Binomial model, the probability of observing th ν > 0 . We willoled DNA . For inh3 in the example, the emission matrix reduces to a row vector of ones, because no data was observed to distinguish among inheritance patterns. In these cases the posterior inheritance probabilities are informed solely by genetically linked loci.For a genotyping platform that produces categorical genotype calls  instead of allele counts, the emission matrix is simply a row vector with a 1 for inheritance patterns that can emit the observed genotype and a 0 for those that cannot. Also in this case, the probability of genotype error could be factored into those values. In case of missing data, such as locus Lpk shown in gdk indicates the imputed marker genotype of DH d at locus k and idk is an incidence vector to indicate the occurrence of the reference allele in the parents of DH d. It contains ones to identify inheritance patterns in pk for which the relevant parent of DH d carries the reference allele and contains zeros where the relevant parent carries the alternate allele.Application of the forward-backward algorithm to the transition and emission matrices for all loci leads to the matrix of posterior inheritance probabilities We numerically evaluated the described approach using Monte-Carlo simulations of scenarios with varying pool sizes, genetic composition and sequencing coverages. We conducted 1,200 independent replications of each simulated scenario to accurately evaluate the expected values of the statistics of interest, which were then summarized in graphical and tabular form . The siIn each replication of the simulation, we generated in-silico 40 biparental Dent by Dent populations, with random selection of the parents from the set of Dent inbred lines. Each line was restricted to use as a parent of only one population. From each population 25 recombinant DH progeny were generated by simulating meiosis between the loci of the parental lines followed by a chromosome doubling step. This was done with the software package ‘hypred’ [Simulation of recombination requires a genetic linkage map of the loci. We obtained this by linear scaling of the physical map positions of the loci to the chromosome lengths of the genetic map reported by Fu et al. . This geA random sample of 200 loci were considered as ‘causal loci’ of a generic polygenic trait. Those markers were subsequently removed from the set of available loci and treated as unobserved. The causal loci were assigned additive substitution effects drawn from a standard Normal distribution. True genetic values for all DH were then calculated by summing the substitution effects according to the genotypes at the corresponding causal loci. To those we then added a Normal distributed noise variable to generate phenotypic values with heritability of 0.5. The genetic and phenotypic values were used only for later application of WGS. They played no role in the imputation process itself.As 35,478 loci carry highly redundant information in F1 derived DH families produced by a single generation of meiosis, we randomly selected a subset of 3,000 of the non-causal SNP loci as genetic markers for genotyping and imputation. This number of markers was previously found to be sufficient for WGS in a collection of biparental populations in maize . All subWe considered pools of two (two-way), three (three-way), and four (four-way) individuals. The pooled individuals either all came from the Dent group (“dent-dent” pools) or from the dent and flint group (“dent-flint” pools). The dent-flint two-way pools comprised one Dent and one Flint individual, three-way pools two Dent and one Flint individual and four-way pools two Dent and two Flint individuals. The pools were formed on a by-population basis, e.g., to form the dent-dent two-way pools, we paired the 25 DH from one Dent population to those of another. To form the four-way pools, we paired the 25 DH from four Dent populations. Within those restrictions, the population pairings and DH pairings within population pairings were chosen at random.To simulate the GBS data of the 3,000 markers for the pooled samples we used the procedure Gorjanc et al.  developeseqk) was sampled from a Gamma distribution with shape and rate of 4 [Sequenceability of each marker  would increase accurate detection of heterogeneous pool genotypes. However, since the goal of the approach is to reduce genotyping costs we considered only low coverage scenarios where the resources consumed by parent-progeny imputation from a pooled sample will be competitive with single-sample GBS.As sequencing coverage levels c et al. , we assux: $6.20, 2x: $5.60, 1x: $5.30, 0.5x: $5.15, 0.25x: $5.08 and, 0.125x: $5.04. To arrive at the genotyping costs per individual, we divided the cost per sample by the number of pooled individuals, assuming that the cost of the pooling step itself was negligible. The ‘true’ marker scores of the DH were treated as obtained from genotyping the individuals separately and at 10x coverage. We will henceforth refer to these as high-quality (HQ) marker scores, in contrast to the marker scores obtained from our pooled genotyping approach, which will be referred to as PG marker scores. The cost per individual for the HQ genotyping was $8.00.To assess the cost efficiency of pooled genotyping with GBS we used the cost model developed by Gorjanc at al.  and avaiThe HMM was applied to the GBS data to obtain imputed reference allele dosages of the pooled individuals for all 3,000 loci. For this we assumed that the parents were genotyped without error  and that all genetic positions are known. Loci monomorphic in all populations contributing to a pool provide no information to linked loci and impute with certainty. To reduce computation time we therefore removed monomorphic loci from the HMM and imputed them directly by filling in the scores of the corresponding parents.For brevity, imputation accuracy was assessed only for the Dent populations. The pools containing Flint populations were used to measure the effect of pooling more genetically diverse individuals than are observed in a dent-dent pool.genotype concordance rate between the true and imputed genotype scores of the polymorphic markers for an individual. We define the genotype concordance rate as the percent of markers for which the genotype with highest posterior probability matches the true genotype. Concordance rates were calculated on a by-individual basis and then summarized by the average and standard deviation across individuals. Those statistics were recorded for each replication of the simulation and then averaged across replications, resulting in numerical evaluations of their expected values.Among the Dent populations, imputation accuracy was measured as the The minimum genotype concordance rate depends upon the allele frequency, so concordance should be interpreted relative to a baseline value obtained by a simple imputation of most frequent genotype . In our multi-polymorphic markers to total polymorphic markers (multi-polymorphism rate) on the genotype concordance rate. We defined multi-polymorphic loci as those polymorphic between the parents of at least one more individual in the pool  will collectively be referred to as “PG-GEBV”.GEBV concordance. Thus, whereas the genotype concordance rate is a direct measure of imputation accuracy, the GEBV concordance can be understood as measuring it indirectly through its effects on WGS. Other studies investigating the use of imputed marker scores for WGS used the correlation between predicted and true genetic values (commonly referred to as the “prediction accuracy”) as indirect measure of imputation accuracy [We measured the impact of imputation accuracy and uncertainty on WGS within the target set by calculating the Pearson correlation between the HQ-GEBV and PG-GEBV of the individuals in the target set. We will refer to this measurement as accuracy , 29. We accuracy . The GEBGenotype imputation is recognized as an accurate and effective way to reduce genotyping costs for WGS in plant breeding , 29, 42.x coverage . For example, a genome-wide π value of 0.6 would represent a 0.6:0.4 ratio of sample DNA contributions in a two-way pool. Differential amplification was absent in our simulations, but it could be measured for each locus from high coverage, non-pooled sequencing of a set of heterozygous individuals [π values. The parameter ν, which specifies the density of the beta-binomial distribution around π, serves to represent general uncertainty in allele dosages when the deviations cannot be estimated empirically for each locus. For example, ν could be increased in an experiment expected to generate a greater degree of allele-specific amplification bias. The Beta-Binomial emission model can also incorporate uncertainty due to residual contamination and other sources of genotyping error. To model genotype error for heterogeneous inheritance patterns, ν can be decreased or increased depending on the amount of genotyping error expected . As expected, the percent of loci that were multi polymorphic was lowest for two-way pools, followed by three-way and four-way pools  and HQ-GEBV of individuals in the prediction set.WGS can be applied within and across populations. Across population selection, however, is largely based on differences in population means , which cx and reached close to 0.95 at 1x coverage, when using the PG marker scores for estimation and prediction. For dent-dent four-way pools the corresponding values were considerably lower at 0.30 and 0.69, respectively.Because PG-GEBV are computed from the PG marker scores, factors affecting the genotype concordance are expected to have a similar effect on the GEBV concordance. Consequently, the GEBV concordance increased with increasing coverage level and was highest for two-way pools followed by three-way and four way pools . For denBecause of the dependence between genotype and GEBV concordance, similar optimization options apply. We found that pooling across germplasm groups led to small but consistent increases in GEBV concordance . We alsoThree main usage scenarios for the PG marker scores can be distinguished: (1) usage for estimation of marker effects only (“estimation only”), (2) usage for calculation of GEBV in the target set only (“prediction only”) and, (3) usage for both (“estimation and prediction”). The GEBV concordance was higher in the “estimation only” scenario than in the “prediction only” scenario . EstimatGEBVs of individuals in the target set, however, are calculated separately for each individual and after marker effects are estimated. Erroneous marker scores then cannot be compensated for by other individuals or linked markers. Prediction is therefore expected to be more sensitive to the errors and uncertainty introduced by usage of PG marker scores. It is therefore even more important for GEBV calculation than marker effect estimation that the genotype uncertainty be incorporated to lower the impact of loci with a greater chance of being incorrect.GEBV concordance was lowest when imputed marker scores were used for both estimation and prediction . This waBecause WGS is most effective when applied to large numbers of genotyped-only individuals , the bulThis leaves the “prediction only” scenario as the most promising option in practice. Here, the increased genotyping efficiency of parent-progeny impuation from pooled samples is applied to where it matters most: the large numbers of unphenotyped individuals in the target set. For those individuals, the marker genotype is the only investment, apart from the relatively minor cost of creating the inbred line through doubled haploidy or repeated selfing . The vasThere is no question that the ability to obtain genotype information of multiple individuals from a single pooled sample considerably decreases genotyping costs. Assuming that the cost of the added DNA pooling step is negligible, genotyping costs would drop two, three, and four-fold, depending on the number of pooled individuals. Additional cost reductions could be achieved by lowering the sequencing coverage level. However, with increasing number of pooled individuals and decreasing sequencing coverage, the GEBV concordance decreases as well, meaning that increased cost savings potential is associated with an increased uncertainty in the calculated GEBV . These tR = irAh, where i is the selection intensity on the auxiliary trait, rA the genetic correlation between the auxiliary and target trait and h is the accuracy with which the auxiliary trait can be assessed [rA is the correlation between true and predicted genetic values. For the HQ marker scores h = 1, because GEBV can be assessed without error. The indirect selection response for HQ-GEBV thus reduces to RHQ = iHQrA. When using PG marker scores, however, the PG-GEBVs themselves are uncertain and so hPG < 1. As estimates of hPG we used the “prediction only” GEBV concordances /iHQ is greater than one.In general, the standardized response to indirect selection is assessed . In the ordances  of the diHQ and iPG are calculated from the fraction of selected individuals s as i = s−1ϕ(Φ−1(1 − s)), where ϕ and Φ are the probability density function and cumulative distribution function of the standard Normal distribution, respectively [sHQ and sPG denote the selected fraction when using HQ or PG marker scores, respectively and let CHQ and CPG be the corresponding costs of genotyping a single individual, as obtained from the previously described cost model. Then sPG = sHQCPG/CHQ, assuming that the same number of individuals is to be selected in each case and the same amount of resources spent on genotyping. In this case, the PG approach can increase the response to selection by increasing the number of individuals from which to select. For example, if a breeder wants to select 10 individuals from a population and can afford to genotype 50 using the HQ marker platform, sHQ would be 0.2. If CPG is just half of CHQ, 100 individuals could be genotyped with the same resources and sPG would equal 0.1. As values of sHQ we choose 0.2 and 0.6. The latter value reflects a scenario in which either WGS is applied only as a pre-test to remove the worst individuals from a population [The selection intensities ectively . Let sHQpulation  or wherex coverage. It then declined again slowly, as any further increase in hPG, which at this point was already above 0.95 , the factor was almost 2.5. This is because the selection intensity i as a function of s follows a curve of diminishing rate of return such that if s is low initially, then a much larger decrease in cost and thereby s is required to affect a sizable increase in i. In cases where the relative increase in response to selection is low, it might be advantageous not to increase the number of genotyped individuals and instead use the freed resources elsewhere. If so, iPG = iHQ and the relative merit of using PG marker scores would be equal to the GEBV concordance. We showed that both the genotype and GEBV concordance can reach very high values already at intermediate GBS coverage levels, meaning that the penalty in selection gain could be minimal. The tremendous cost savings potential of pooled genotyping could then benefit those components of the breeding operation where the return on investment is greatest.The relative merit of PG over HQ increased on average with increasing coverage level . For denove 0.95 , could nhed 0.95 . The optTo summarize, in this study we presented a method for parent-progeny imputation from pooled samples and applied it to simulated GBS data from biparental populations. We demonstrated that the imputed genotypes can be very accurate even at low coverage levels and then only minimally affect the estimation of marker effects or calculation of GEBV in WGS. The tremendous cost savings potential of the method can therefore facilitate large scale genotyping in plant breeding, a key requirement for successful applications of WGS.S1 File1 and D1, respectively. Those of the second DH are R2 and D2. The recombination frequency between locus k and k − 1 is rk.The recurrent and donor parents of the first DH are R(PDF)Click here for additional data file.S2 File1, L2, L3 and L4 for a DNA pool of three DH individuals  from three biparental populations .Parent-progeny imputation is carried out for four genetically linked loci L(PDF)Click here for additional data file.S3 FileZipped folder with R code and instructions for generating the simulated data as described in Materials and Methods .(ZIP)Click here for additional data file.S1 Fig(PDF)Click here for additional data file.S2 Figx. Those results are replicated here for comparison purposes.The percent of missing markers (in parentheses) correspond to the expectations at the indicated sequencing coverage levels. The full line shows results from the standard GBS scenario where the read number and % missing loci varies as a function of the sequencing coverage (PDF)Click here for additional data file.S3 Fig(PDF)Click here for additional data file.S4 Fig(PDF)Click here for additional data file.S1 Table(XLSX)Click here for additional data file.