We have recently developed analysis methods (GREML) to estimate the genetic variance of a complex trait/disease and the genetic correlation between two complex traits/diseases using genome-wide single nucleotide polymorphism (SNP) data in unrelated individuals. Here we use analytical derivations and simulations to quantify the sampling variance of the estimate of the proportion of phenotypic variance captured by all SNPs for quantitative traits and case-control studies. We also derive the approximate sampling variance of the estimate of a genetic correlation in a bivariate analysis, when two complex traits are either measured on the same or different individuals. We show that the sampling variance is inversely proportional to the number of pairwise contrasts in the analysis and to the variance in SNP-derived genetic relationships. For bivariate analysis, the sampling variance of the genetic correlation additionally depends on the harmonic mean of the proportion of variance explained by the SNPs for the two traits and the genetic correlation between the traits, and depends on the phenotypic correlation when the traits are measured on the same individuals. We provide an online tool for calculating the power of detecting genetic (co)variation using genome-wide SNP data. The new theory and online tool will be helpful to plan experimental designs to estimate the missing heritability that has not yet been fully revealed through genome-wide association studies, and to estimate the genetic overlap between complex traits (diseases) in particular when the traits (diseases) are not measured on the same samples. Genome-wide association studies (GWAS) have identified thousands of genetic variants for hundreds of traits and diseases. However, the genetic variants discovered from GWAS only explained a small fraction of the heritability, resulting in the question of “missing heritability”. We have recently developed approaches  to estimate the overall contribution of all SNPs to the phenotypic variance of a trait (disease) and the proportion of genetic overlap between traits (diseases). A frequently asked question is that how many samples are required to estimate the proportion of variance attributable to all SNPs and the proportion of genetic overlap with useful precision. In this study, we derive the standard errors of the estimated parameters from theory and find that they are highly consistent with those observed values from published results and those obtained from simulation. The theory together with an online application tool will be helpful to plan experimental design to quantify the missing heritability, and to estimate the genetic overlap between traits (diseases) especially when it is unfeasible to have the traits (diseases) measured on the same individuals. Therefore, if there are many genes each with a small effect affecting the trait, most of these genetic variants will fail to pass the stringent threshold and remain undetected. This is one of the explanations of the ‘missing heritability’ question, that genetic variants identified from GWAS so far explain a fraction of the heritability for complex traits Genome-wide association studies (GWAS) have been extremely successfully in identifying genetic variants associated with complex traits and diseases in humans We previously named the SNP-based method mentioned GREML y is an N×1 vector of phenotypes with N being the sample size, g is an N×1 vector with each of its elements being the total genetic effect of an individual captured by all SNPs, and e is an N×1 vector of residuals. We have A is the genetic relationship matrix (GRM) estimated from SNPs I is an identity matrix. The genetic relationships, also known as ‘genomic relationships’ or ‘genetic similarity relationships’, are referenced to the current population, and so can be negative as they are distributed about a mean of zero. The methods of using SNP data to estimate genetic variance in unrelated individuals have been detailed elsewhere For power calculation, we need to know the sampling variance of the estimate of i and j  or identity-by-state (IBS) at genome-wide markers ype data  that thiype data . The SEsy1 and y2 are N×1 vector of phenotypes, g1 and g2 are N×1 vectors of genetic effects with e1 and e2 are N×1 vectors of residuals with N is the sample size. The variance covariance matrix isFor a bivariate analysis where the two traits are measured on the same individuals, the mixed linear model can be written as  problem , i.e.(7)y1 is an N1×1 vector of phenotypes in sample set #1 , and y2 is an N2×1 vector of phenotypes for in sample set #2 , with N1 and N2 being the sample sizes of the two sets. A1 is an N1×N1 GRM for individuals in sample set #1, A2 is an N2×N2 GRM in sample set #2 and A12 is an N1×N2 GRM between the two sets of samples. I1 and I2. For a bivariate analysis where the two traits are measured on different sets of individuals, e.g. height in males and blood pressure in females, the variance-covariance matrix is i-th individual in set #1 and the j-th individual in set #2, and ij-th element of the GRM A12, i.e. the genetic relationship between the i-th individual in set 1 and the j-th individual in sample set #2. The regression coefficient is equivalent to genetic covariance between the two traits becauseAnalogous to the univariate analysis, estimation of genetic covariance by a bivariate mixed linear model analysis is asymptotically equivalent to the following linear regression modelWe know from the derivations above that ulations  and the ulations . When , N being the total number of cases and controls. We show in For case-control studies, the proportion of variance in case-control status (0 or 1) that is explained by all SNPs on the observed scale (K), the proportion of variance in disease liability that is captured by the SNPs at population level, and the proportional of cases in the sample (v). For this reason we define Ncase is the number of cases, Ncontrol is the number of controls, and i is the selection intensity which is a function of KK) and proportion of cases in the sample (v) due to the transformation.To calculate power, however, we would need to specify N1 and N2 are the total numbers of cases and controls of the two case-control studies, respectively. This also applies to a bivariate analysis of a quantitative trait and a cases-control disease study on different sets of samples, i.e.As shown in θ, where θ is either the proportion of phenotypic variance captured by SNPs  of λ and the type-I error rate of α, statistical power is the probability that a non-central χ2 variable is larger than the central χ2 threshold that is determined by α. We show in Statistical power is calculated from the population value of the parameter and its sampling variance, which was derived above. If the parameter is http://spark.rstudio.com/ctgg/gctaPower), as part of the GCTA http://ctgg.qbi.uq.edu.au/software/gcta), using R-Shiny (http://shiny.rstudio.org) to calculate the SE of genetic variance or genetic correlation and statistical power given user-defined parameters.We have also developed an online calculator  and the variance in SNP-based genetic relatedness variance from pedigree analyses have been around for over 50 years L is the likelihood function, and V matrix can be computed explicitly, so that the NCP can be calculated without making approximation Methods for calculating the power of detecting quantitative trait loci (QTL) in family-based linkage studies have been investigated extensively in the past two decades For a given population, a set of common SNPs and the method of calculating the genetic relationship matrix that we have used here, If there are unknown related samples in the data (cryptic relatedness), Using the same experimental design of a sample of conventionally unrelated individuals, the experimenter can increase power by increasing sample size. Fortunately, power increases quadratically with sample size because every new sample is contrasted with all existing samples. The sampling variance of the estimate of the genetic correlation is generally much larger than that of the proportion of variance explained from a univariate analysis, consistent with the theory of the sampling variance of genetic correlations in pedigree designs Figure S1Likelihood ratio test (LRT) statistic vs. Chi-squared test-statistic in a univariate analysis.(PDF)Click here for additional data file.Table S1Standard error of the estimate of (PDF)Click here for additional data file.Text S1Simulations.(PDF)Click here for additional data file.Text S2Sampling variance of genetic correlation.(PDF)Click here for additional data file.Text S3Acknowledgments to dbGaP data.(PDF)Click here for additional data file.


Triticum aestivum L.) lines and hybrids adapted to Central Europe in field trials in Germany and fingerprinted the lines with a 9000 single-nucleotide polymorphism array. Additive effects prevailed over dominance effects. A two-dimensional genome scan revealed the presence of epistatic effects. Genome-wide association mapping in combination with a robust cross-validation strategy identified one frost tolerance locus with a major effect located on chromosome 5B. This locus was not in linkage disequilibrium with the known frost loci Fr-B1 and Fr-B2. The use of the detected diagnostic markers on chromosome 5B, however, does not allow prediction of frost tolerance with high accuracy. Application of genome-wide selection approaches that take into account also loci with small effect sizes considerably improved prediction of the genetic variation of frost tolerance in wheat. The developed prediction model is valuable for improving frost tolerance because this trait displays a wide variation in occurrence across years and is therefore a difficult target for conventional phenotypic selection.Abiotic stress tolerance in plants is pivotal to increase yield stability, but its genetic basis is still poorly understood. To gain insight into the genetic architecture of frost tolerance, this work evaluated a large mapping population of 1739 wheat ( Triticum aestivum L.) is one of the most important crops and is grown on 200 million hectares of farmland worldwide  array . For hybrid seed production, the 135 parental inbred lines were classified into a male group consisting of tall and open-pollinating genotypes and a female group of semi-dwarf genotypes showing a short delay in flowering time compared to the male lines. The hybrids were derived by crossing 120 female and 15 male lines  using chemical hybridization agents. The 1739 genotypes were evaluated for frost tolerance in field trials at three locations in Germany  in the year 2012. The experimental designs were partially replicated alpha designs and 29% of the hybrids were tested in two replications . Sowing was done in the first 2 weeks of October and sowing densities ranged from 230 to 290 grains m–2 and plot sizes ranged from 7.5 to 9.7 m2. The plots were treated with fertilizers, fungicides, and herbicides according to standard agronomic practices for intensive wheat production. Frost tolerance was visually scored on a scale from 1 (no damage) to 9 (no plant survived).This study was based on 1604 elite wheat hybrids and their 135 parental inbred lines adapted to Central Europe (Supplementary Table S2). This work followed the suggestion of θij between inbreds i and j on the basis of marker data as θij = 1 + (Sij – 1)/(1 – T), where Sij is the proportion of marker loci with shared variants between inbreds i and j and T is the average probability that a variant from one parent of inbred i and a variant from one parent of inbred j are alike in state, given that they are not identical by descent. T was set as minimum of (1 – Sij) values.Genotyping was done with the 9K SNP array based on the Illumina Infinium assay  concomitant with an absence of snow coverage and soil-borne diseases. The high correlations observed among the genotypic values estimated in the three locations (Supplementary Table S3) clearly suggested that a combined analysis across locations would not be biased by location-specific frost responses. The high frost stress revealed a large genetic variation for frost tolerance in the mapping population with high heritabilities . Julius is known to be frost tolerant and Tabasco frost susceptible.The 1739 entries were evaluated for frost tolerance occurring in the field under natural settings. Frost stress was severe at all three locations in the first half of February 2012, with minimum temperature down to –18 °C . This can be explained by a steady interchange of lines among different wheat breeding programmes in their breeding history , population stratification had to be considered through the kinship matrix for the lines and hybrids for a single-locus regression model . Fitting an average heterotic effect, however, was not required. Mapping QTL in a combined population of hybrids and lines led to a substantially higher proportion of genotypic variance explained by the QTL . This held true even if the proportion of genotypic variation explained by SNPs was estimated separately for the parental lines and the hybrids. Therefore, the genome-wide mapping approaches were based on the combined population of hybrids and lines.Based on the genome-wide SNP marker data, there was no major population structure, but the presence of a family structure among the 135 parental lines (P < 0.01) interaction effects with the environments. A full two-dimensional scan for epistatic effects was performed, which revealed a total of 76 significant digenic epistatic effects. The distribution of the P-values revealed that, among the four types of digenic epistatic effects, additive × additive interactions were the most prevailing .In the genome-wide association mapping scan, there were 26 and 15 SNPs which contributed significantly to the additive and dominance genetic variation for frost tolerance, respectively . About 7Supplementary Table S4) was sufficient to explain 98% of the prediction accuracy of the total set of significant SNPs. These five SNPs all exhibit only significant additive effects, are located in the same genomic region on chromosome 5B, and have r2 values amongst each other above 0.88.Using a robust cross-validation strategy in which training and validation sets were not linked through common parents, the phenotypic performance of unrelated hybrids could be predicted with an accuracy of 0.34 based on the SNPs detected in the association mapping scans . By stanFr-B1/Vrn-B1, which is located roughly in the middle of the long arm  genes at the Fr-B2 locus, also located on the long arm of chromosome 5B. The current work used the developed PCR markers for CBF-B9 and CBF-B14  compared to the association mapping approach. BayesCπ and RR-BLUP showed negligible differences in the accuracy to predict frost tolerance despite the low posterior inclusion of markers in BayesCπ (20% additive and 6% dominance). There was an accuracy of prediction of frost tolerance with dominance effects based on RR-BLUP and BayesCπ of around 0.20. Nevertheless, a combined prediction considering additive and dominance effects was not superior to predicting frost tolerance exclusively by additive effects.r2 values among adjacent loci of above 0.2 is useful for genome-wide approaches to predict phenotypic performance. The current study’s companion paper (r2) between adjacent SNPs of 0.32, which suggests that genome-wide mapping is promising with this experimental set up.A crucial factor in determining the quality of molecular data for genome-wide mapping studies is the extent of linkage disequilibrium. Theoretical findings  between adjacent SNPs. Obviously the mapping resolution can still profit from the 10% of variation not explained by adjacent SNPs. The alleles of the SNPs detected in the scan on chromosome 5B were not in linkage disequilibrium with known polymorphisms at the Fr-B1 and Fr-B2 loci. Consequently, these findings suggested that a new frost stress locus had been detected on chromosome 5B.The genome-wide association mapping scan in combination with the robust cross-validation approach suggests that one gene with a major effect is present roughly in the middle of the long arm of chromosome 5B . InteresVrn-A1 contributes substantially to the genetic variation of frost tolerance when differences are tested between winter and spring alleles  and skewness of P-values towards zero . This predominance of additive × additive effects is expected for selfing species such as wheat . Despite these small effect sizes, the large number of potential interaction effects can lead to a substantial contribution of epistasis to the phenotypic variation of frost tolerance. Consequently, the presence of epistatic effects is an explanation for this study’s inability to predict frost tolerance with a higher accuracy than observed.The observed accuracy of prediction of frost tolerance  suggestsas wheat . The ideThis study provides insights into the genetic architecture of frost tolerance in wheat adapted to Central Europe. One major QTL on chromosome 5B segregating in European winter wheat was detected, which is of use for marker-assisted selection after thorough validation in independent experiments. In addition, for the first time it has been shown that using genomic selection approaches that assume a more complex genetic architecture represents a major step to improve understanding of the genetic architecture of frost tolerance. The developed calibration models are valuable to improve the frost tolerance of wheat, which is a trait that displays a wide variation in occurrence across years and is therefore a difficult target for conventional phenotypic selection.JXB online.Supplementary data are available at Supplementary Fig. S1. Crossing scheme between the 120 female and 15 male parental wheat linesSupplementary Fig. S2. Temperature profiles of the three locationsSupplementary Fig. S3. Distribution of the phenotypic values of 1739 genotypes for frost toleranceSupplementary Fig. S4. Coefficients of parentage estimated for the 120 female and 15 male parental wheat lines from SNP marker dataSupplementary Fig. S5. Quantile–quantile plots for association mapping based on the combined population of hybrids and parental lines using four different biometrical approachesSupplementary Fig. S6. Distribution of P-values for four different types of digenic epistatic effectsSupplementary Fig. S7. Box-whisker plots of the proportion of explained phenotypic variation for significant digenic epistatic effectsSupplementary Table S1. Description of the 135 parental lines used in this studySupplementary Table S2. SNP data of the 135 parental linesSupplementary Table S3. Phenotypic correlations of the adjusted entry means of 1604 hybrids and 135 lines of three environmentsSupplementary Table S4. Summary of marker statistics for the three applied genome-wide mapping approaches


N = 1596). General cognitive ability was assessed at age 11 years and in late life  and was modelled against the total number of stop-gain/loss, splice, and missense exonic variants, with minor allele frequency less than or equal to 0.01, using linear regression adjusted for age and sex. In both cohorts and in both the childhood and late-life models, there were no significant associations between rare variant burden in the exome and cognitive ability that survived correction for multiple testing. Contrary to our a priori hypothesis, we observed no evidence for an association between the total number of rare exonic variants and either childhood cognitive ability or late-life cognitive ability.Human cognitive ability shows consistent, positive associations with fitness components across the life-course. Underlying genetic variation should therefore be depleted by selection, which is not observed. Genetic variation in general cognitive ability (intelligence) could be maintained by a mutation–selection balance, with rare variants contributing to its genetic architecture. This study examines the association between the total number of rare stop-gain/loss, splice and missense exonic variants and cognitive ability in childhood and old age in the same individuals. Exome array data were obtained in the Lothian Birth Cohorts of 1921 and 1936 (combined  This mag –5. Twin,50–80%) . At the ity rate , better 0% . At tity rate , height ity rate , developity rate ,12, sperity rate  and highity rate  and occuity rate . Furthere choice –18. Thisansition ,20. Geneansition . Cognitiansition ,22,23. Fansition ,12,24, aansition , suggestansition . These lansition , i.e. viansition .g in the range of 30–50% [g consists either of common genetic variants with individually miniscule phenotypic effects, or of rare genetic variants that have been in the population for long enough that they are in sufficient linkage disequilibrium with the common SNP markers present on commercial genotyping arrays. More recent mutations, including family-specific and private de novo genetic variants, could explain the discrepancy in heritability estimates from GCTA compared to those from twin, family and adoption studies [Since selection will quickly purge genetic variants with strong effects on fitness-related traits, genetic variation maintained by mutation–selection balance suggests a genetic architecture that lacks common genetic variants of notable effect sizes ,28. Thisf 30–50% . Since t studies .g, while the individual causal variants might be heterogeneous and interchangeable [et al. [g. The recently developed exome array (http://genome.sph.umich.edu/wiki/Exome_Chip_Design), which genotypes rare variants identified from the exome sequencing of around 12 000 persons, provides an ideal resource to test this hypothesis.A genetic architecture based on rare genetic variants implies high genetic heterogeneity ,28 and mangeable . Using a [et al.  found a  [et al. –36. A diWhile the rank-order stability of cognitive ability is high even from childhood to old age ,38, cognThe aim of this study was to examine the relationship between an individual's total number of rare exonic variants (SNPs with MAF ≤ 0.01) with cognitive ability measured in the same individuals at two points in life, during childhood and old age.2.(a)Data stem from two Scottish birth cohorts: the Lothian Birth Cohorts (LBC) of 1921 (LBC1921) and 1936 (LBC1936), which are longitudinal studies of ageing –44.LBC1921 participants were born in 1921 and most completed the Scottish Mental Survey 1932 (SMS1932) at mean age 11 years . The SMSThe design of LBC1936 is similar to LBC1921. Participants were born in 1936 and completed the SMS1947, which was administered nationwide to all children born in that year who attended school in Scotland on 4 June 1947 . The Mor(b)Following informed consent, venesected whole blood was collected for DNA extraction in both LBC1921 and LBC1936. Ethics permission for the LBC1921 was obtained from the Lothian Research Ethics Committee  and the Scotland A Research Ethics Committee (wave 4: 10/S1103/6). Ethics permission for the LBC1936 was obtained from the Multi-Centre Research Ethics Committee for Scotland (wave 1: MREC/01/0/56), the Lothian Research Ethics Committee (wave 1: LREC/2003/2/29) and the Scotland A Research Ethics Committee (waves 2 and 3: 07/MRE00/58). Written informed consent was obtained from all subjects.(c)z-standardized scores from the first principal component, which follow a Gaussian distribution with mean 0, variance 1, were saved and used as the measure of late-life cognitive ability.For the current analysis, we consider two measures of cognitive ability. The same measure of general cognitive ability, the Moray House Test ,46, was (d)n = 3590 in LBC1921, n = 8248 in LBC1936). Monomorphic SNPs , SNPs with a call rate less than 97% , and individuals with a call rate less than 97%  were removed. This left a total of 72 016 SNPs for downstream analysis in LBC1921 and 82 845 SNPs in LBC1936.Genotype data were collected on the Illumina HumanExome genotyping array. Initial quality control was performed according to the CHARGE consortium criteria , leaving(e)n = 36 200 in LBC1921, n = 46 384 in LBC1936). The global burden of rare variants for each individual was defined as the total number of rare variant alleles (an additive model).From the cleaned exome array data, stop-gain/loss, splice and missense SNPs with a MAF of less than or equal to 0.01 were extracted ((f)n = 2 in LBC1921, n = 3 in LBC1936). All models were re-run as part of a sensitivity analysis that used a less stringent cut-off for rare variants . Models were also run at both the 1% and 5% MAF threshold to analyse individually the effects of rare variant burden on cognition for (i) stop-gain/loss, (ii) splice, (iii) missense and (iv) all variants, including synonymous SNPs. Diagnostic plots were examined to check model fit. All analyses were conducted in R [Linear regression models were built to assess the relationship between rare variant burden and childhood and late-life cognition in LBC1921 and LBC1936. Mean-centred age at cognitive assessment and sex were included as covariates in each model. The global burden of rare variants followed a Gaussian distribution; however, a few extreme outliers were removed prior to analysis  and 49.0  in LBC1921 and LBC1936, respectively. The corresponding values in late life were 59.4  and 64.3 . The mean numbers of rare variants per individual were 185.9  in LBC1921 and 184.7  in LBC1936.b = −0.047 , p = 0.17; LBC1936: b = 1.2 × 10−3 , p = 0.96). Similar null findings were observed between burden and late-life Moray House Test scores , p = 0.78; LBC1936: b = −0.016 , p = 0.35). There were also null findings for the late-life models of burden against fluid general cognitive ability , p = 0.62; LBC1936: b = 2.1 × 10−3 , p = 0.26), which are presented in b = 0.23, p = 0.84 in LBC1921, b = 1.75, p = 0.021 in LBC1936) and slightly lower scores in later life . A similar pattern was seen for the late-life fluid abilities . Older children performed better on the Moray House Test at age 11 by around eight points per year (p < 0.001) but older adults had lower Moray House Test  and g scores  in late life.Some age and sex effects were observed in both early and late life and for both general and fluid cognitive ability tables  and 3. Fp = 0.039 and 0.011, respectively). However, these associations were not significant after a Bonferroni correction to account for multiple testing.Sensitivity analyses that included SNPs with a MAF less than 0.05 yielded near identical null associations to the models with a MAF cut-off of 0.01 . There were no associations between burden and cognitive ability for the models that considered (i) stop-gain/loss, (ii) splice, (iii) missense and (iv) all variants, including synonymous SNPs . The only exception was for the splice variants in LBC1936, where a unit increase in rare variant burden decreased the childhood and late-life MHT scores by 0.3 points  for the present analysis although re-adjusting this to 0.05 in a secondary analysis made no difference in the findings.et al. [A possible explanation for our null finding might be that rare non-synonymous exonic SNPs, which make up the bulk (approx. 96%) of our variants, are not the right place to look for genetic variants explaining normal variation in cognitive ability. Non-synonymous exonic variants alter protein structures, which might result in too radical phenotypic effects to be observed in the normal, healthy population range. A sensitivity analysis that considered the burden of both synonymous and stop-gain/loss, splice and missense variants made very little difference to the association between total burden score cognitive ability. Exonic point mutations have been found to be mostly recent , which wet al.  found deet al.  and assoet al. –60, an iet al. ,62, suggAnalyses such as the one conducted in this paper may also be better applied to complex traits with dichotomous end-points  e.g. Alza priori hypothesis, we observed no evidence for an association between the total number of rare exonic variants and cognitive ability in childhood and old age. Future studies await the availability of whole-genome sequencing data in order to test associations of rare variant burden in the whole genome. These studies should also consider additional measures of fitness to see whether our findings are more widely applicable to other traits that can be considered being under mutation–selection balance. Ideally, these analyses should be conducted on populations where a full spectrum of the fitness measure is represented or where there is a well-defined dichotomous split in the trait.In conclusion, contrary to our


Statistical genetics can be viewed as a classical branch of applied probability and statistics, which has recently gained much momentum, due to the significant breakthroughs in genetics. With the availability of modern techniques, new methods, and significantly increased data information, it is imperative to study the relationship between gene traits such as diseases and genetic susceptibilities in an unprecedented manner. This area is among the hottest topics in applied statistics, applied mathematics, biological/medical studies, and other related sciences.The main aim of this special issue focuses on the new development and applications of computational, mathematical, and statistical methods in genetic disease study. The special issue could become an international forum for researchers to exchange their new thoughts, most recent developments, and ideas in the field.In this special issue, we selected seven articles within the above topics. Below is a brief summary of the selected articles in this special issue.Applications of bayesian gene selection and classification with mixtures of generalized singular g-priors” by W.-K. Chien and C. K. Hsiao addresses this problem using Bayesian method with a Gaussian prior and inverse gamma hyperprior. The proposed approach is applied to a colon and leukemia cancer study. Comparison with other existing methods was conducted. The authors find that classification accuracy of the proposed model is higher with a smaller set of selected genes and that the results not only replicated findings in several earlier studies, but also provided the strength of association with posterior probabilities.Recent advances in biotechnologies have led to the identification of an enormous number of genetic markers in disease association studies; how to select a smaller set of genes to explore the relation between genes and disease is a challenging task. Bayesian methods have the advantage of incorporating prior information into the model for such analysis. Article “Modified logistic regression models using gene coexpression and clinical features to predict prostate cancer progression” by H. Zhao et al. proposed a new logistic regression model for predicting prostate cancer progression. They incorporated coexpressed gene profiles into the logistic model based on clinical features to improve the inference accuracy. Then they use the top-scoring pair method to select genes with significant association with the disease. The performance of the proposed method is compared with some commonly used methods for such problem, using data sets from such published studies. Their study suggests that the proposed method performs better than a commonly used one and that the top-scoring pair method is a useful tool for feature (and/or gene) selection to be used in prognostic models.Article “Power and stability properties of resampling-based multiple testing procedures with applications to gene oncology studies” by D. Li and T. D. Dye investigates the power and stability of seven commonly used resampling-based multiple testing procedures that are frequently used in high-throughput data analysis for small sample size data. Simulations and real data gene oncology examples are employed in their investigation. Their study suggests that the bootstrap single-step minP procedure and the bootstrap step-down minP procedure perform the best, when sample size is as small as 3 in each group and either familywise error rate or false discovery rate control is desired. When sample size increases to 12 and false discovery rate control is desired, the permutation maxT procedure and the permutation minP procedure perform the best.Resampling-based multiple testing procedures are widely used in genomic studies to identify differentially expressed genes and for genome-wide association studies. The power and stability of these popular procedures have not been extensively evaluated. Article “Transcriptional protein-protein cooperativity in POU/HMG/DNA complexes revealed by normal mode analysis” by D. D. Wang and H. Yan investigates how proteins in POU/HMG/DNA ternary complexes interact cooperatively, which are crucial in transcriptional regulation of embryonic stem cells. They use the normal mode analysis to detect the most cooperative or collective motions  of a large number of proteins, a commonly used tool to analyze the structural dynamics of biomolecules, which combines some techniques in engineering, mathematics, and statistics. Their work reveals how the two proteins Oct-1 and Sox-2 work together physically and structurally at two specific DNA biding sites, by analyzing the motion magnitude functions. A correlation measure is used to characterize the amount of cooperativity of pairs of proteins. The proposed methods provide useful information for understanding the complicated interaction mechanism in the POU/HMG/DNA complexes. The corresponding online computational tools are also provided.Article “Variable selection in ROC regression” by B. Wang addresses this problem with an interesting idea. There are a large number of criteria available for this problem. The author first rewrites the ROC regression into a grouped variable selection form so that current criteria can be applied and then proposes a general two-stage framework with a BIC selector for the group SCAD algorithm under the local model assumption. Basic asymptotic properties of the proposed methods are derived. Simulation studies and real data analysis show that the proposed grouped variable selection is superior to the traditional model selections. Furthermore, the author finds that the focused information criterion provides more accurate estimated area under the ROC curve compared with other criteria.In modern medical diagnosis or genetic studies, the receiver operating characteristic (ROC) curve is a popular tool to evaluate the discrimination performance of biomarkers on a disease status or a phenotype. With the presence of a number of covariates in the data, how to select the most relevant covariables, or to select the model with good overall properties, is a challenging problem. Article “Robust joint analysis with data fusion in two-stage quantitative trait genome-wide association studies” by D.-D. Pan et al. proposes a powerful F-statistic-based robust joint analysis method for quantitative traits using the combined raw data from both stages, in which the genetic effects are modeled as regression parameters. Variations of the MAX testing statistic are constructed to calculate the statistical significance and power. It is well known that critical values and power of the MAX type statistic are not easy to compute. The authors derived analytic expressions on the basis of the asymptotic distributions, so that these quantities can be easily obtained. They show using simulations that the proposed method is substantially more robust than the F-test based on the commonly used additive model when the underlying genetic model is unknown.Two-stage design and analysis are often adopted in genome-wide association studies (GWASs). Considering the genetic model uncertainty, many robust procedures have been proposed and applied in GWASs. The existing approaches mostly focused on binary traits, and many of these methods analyze data based on two separate stages, and few work has been done on continuous (quantitative) traits. Article “A statistical method for synthesizing meta-analyses” by L. L. Tang et al. introduces a method to synthesize the meta-analytic results under two cases: (1) when multiple meta-analyses use the same type of summary effect estimates and (2) when meta-analyses use different types of effect sizes. In case 2, the meta-analysis results cannot be directly combined; therefore they propose a two-step frequentist procedure to first convert the effect size estimates to the same metric and then summarize them with a weighted mean estimate. The proposed method has the following advantages over some existing methods: different types of summary effect sizes can be considered; the same overall effect size can be provided by conducting a meta-analysis on all individual studies from multiple meta-analyses.Multiple meta-analyses may use similar search criteria and focus on the same topic of interest, but they may yield different or sometimes discordant results. The lack of statistical methods for synthesizing these findings makes it challenging to properly interpret the results from multiple meta-analyses, especially when their results are conflicting. Article “SNP selection in genome-wide association studies via penalized support vector machine with MAX test” by J. Kim et al. proposes a two-stage method such that the genetic models of each SNP are identified using the MAX test and then a prediction model is fitted using a penalized SVM method. They apply the proposed method to various penalized SVMs and compare their performances using various penalty functions. They show by simulations and real GWAS data analysis that the proposed method performs better than the prediction methods that ignore the genetic models, in terms of prediction power and selectivity.One of the main objectives of a genome-wide association study (GWAS) is to develop a prediction model for a binary clinical outcome using single-nucleotide polymorphisms (SNPs) which can be used for diagnostic and prognostic purposes and for better understanding of the relationship between the disease and SNPs. Penalized support vector machine (SVM) methods have been widely used toward this end. However, since investigators often ignore the genetic models of SNPs, a final model results in a loss of efficiency in prediction of the clinical outcome. Article “On coalescence analysis using genealogy rooted trees” by A. Yuan et al. explores a new method for this problem. They first investigate the asymptotic behavior of such inference; their results indicate that, broadly, the estimated coalescent time will be consistent to a finite limit. Then they study a relatively simple computation method for this analysis and illustrate how to use it.Using DNA sequence data in the study of ancestral history of human population is an essential part in the understanding of human evolution. The existing methods for such coalescence inference using the method of either the rooted tree or unrooted tree constructed from the observed data, both of which use recursion formulae to compute the data probabilities. These methods are useful in practical applications but computationally complicated. Article “


Genomic imprinting is an important epigenetic phenomenon, which on the phenotypic level can be detected by the difference between the two heterozygote classes of a gene. Imprinted genes are important in both the development of the placenta and the embryo, and we hypothesized that imprinted genes might be involved in female fertility traits. We therefore performed an association study for imprinted genes related to female fertility traits in two commercial pig populations. For this purpose, 309 SNPs in fifteen evolutionary conserved imprinted regions were genotyped on 689 and 1050 pigs from the two pig populations. A single SNP association study was used to detect additive, dominant and imprinting effects related to four reproduction traits; total number of piglets born, the number of piglets born alive, the total weight of the piglets born and the total weight of the piglets born alive. Several SNPs showed significant ( Genomic imprinting is an epigenetic phenomenon where the degree of expression of an allele depends on its parental origin. The parent-of-origin-dependent allele expression of genomically imprinted genes is controlled by epigenetic marks such as DNA methylation and histone modifications which are established during gametogenesis and mostly maintained during life http://igc.otago.ac.nz and http://www.geneimprint.com/site/genes-by-species), several hundreds of genes have been predicted to be imprinted in human and mouse Genomic imprinting has been found in viviparous mammals and in seeded plants The majority of genomically imprinted genes are found in clusters containing protein coding and non-coding genes http://igc.otago.ac.nz for an overview).At the phenotypic level, imprinting is manifested through a contrast between the two heterozygote classes that exist for a genotype  A common denominator in genome screens for imprinted QTL in pigs is the use of experimental crosses between divergent pig breeds or lines. When the lines are not completely inbred, this incurs the risk of false positive detection of imprinted QTL due to heterogeneity in the original purebred populations One of the most intensively studied imprinted QTL in pigs is the paternally expressed QTL on chromosome 2, which affects heart muscle size, muscle growth and fat deposition Several hypotheses for the evolution of genomic imprinting have been formulated, many related to allocation of resources from mother to offspring during the early stages of development. These hypotheses include: the parental conflict hypothesis that explains genomic imprinting by a parental conflict in allocation of resources to the offspring The presumption that genomically imprinted genes regulate the resource allocation between mother and offspring The aim of this study was therefore to explore whether putative imprinted genes or regions associate with fertility traits in commercial pigs. For this purpose, fifteen evolutionary conserved imprinted regions were genotyped in two commercial pig breeds. An association study was used to detect additive, dominant and imprinting effects related to four reproduction traits , the number of piglets born alive (LB), the total weight of the piglets born (TW) and the total weight of the piglets born alive (LW)). Several additive and dominant associations and one imprinted association were detected. These results are discussed in relation to their biological relevance.The data of two commercial purebred pig populations were analyzed in this study. Both populations were Large White dam lines which have been selected for several generations for commercially important traits, including reproduction traits. The traits analyzed in this study were reproductive performance of the sows, based on their litters. Some of the litters were purebred and others were crossbreds. Phenotypes considered were the total number of piglets born (TB), the number of piglets born alive (LB), the total weight of the piglets born (TW) and the total weight of the piglets born alive (LW).  overlap The fifteen selected regions are located on ten different chromosomes with three regions on chromosome 1, two regions on chromosomes 2, 9, and 17 and one region on chromosomes 5, 6, 7, 8, 14 and 18 . The sizSingle SNP association analyses were performed to detect additive, dominance and imprinting effects related to the four traits. For each combination of trait and population, several additive, dominant and imprinted effects had a  see . The p-vulations . In popuulations . The absr traits .The imprinting effect in population C1 with significant FDR in region 7_1 on trait TB corresponded to SNP marker ASGA0037226. In this population, this region contained several other markers with small The significant imprinting effect in region 7_1 on trait TB in population C1 explained 1.6% of the phenotypic variance of trait TB , which rEstimates for LD in region 7_1  revealedTo ensure that the observed imprinted effect was not an effect of a stochastic unequally assignment of parental alleles from heterozygotic parents, genotypic means were also calculated based on matings that resulted in irrefutable allele origin in the offspring (e.g a BA genotype from a AA mother and a BB father). In both populations, the means for LB and TB of the BA genotype where higher than those of the AB genotype, validating the imprinting effect (results not shown). The deviation from the expected Hardy-Weinberg equilibrium can be specific for the sampled populations and therefore we also estimated these deviations for the other markers. For this purpose, the Fertility is an economically important trait in the pig breeding industry for which considerable selection has been applied in the last decades. Many studies have been conducted to find QTL and genes related to reproduction traits in pigs  was calculated. A consequence was that we used a stringent significance thresholds for our tests, leading to reduced power to detect imprinting effect, but strengthening the confidence in the detected effects. The fact that we only found significant evidence for one imprinted effect is partially due to this reduced power, but does also illustrate the challenge of detecting imprinted effects in association studies.The proportion of phenotypic variance explained by this imprinted effect was substantial, accounting for 1.6% of the phenotypic variance . In population C2, the imprinting effect of this marker was not significant, but the estimated imprinting effect had the same sign as in population C1 .We performed additional analyses using haplotypes instead of single SNP and fitting additive, dominance and imprinting effects as random effects. Results from this analysis show that the variance explained by imprinting effects was approximately equal to the imprinting variance based on the single SNP analysis. These results suggest that the SNP ASGA0037226 is in weak LD with other SNPs in this region and that the association between the QTL and these other SNPs is weak. This is in line with the LD pattern in region 7_1 Region 7_1 corresponds to the DLK1-DIO3 imprinted domain which contains at least three maternal imprinted protein coding genes  and many paternal imprinted small and large ncRNA genes. The SNP marker with significant imprinted effect (ASGA0037226) is located approximately 25 kb from the DIO3 gene and about 500 kb from other known imprinted genes in this region. DIO3 codes for type 3 deiodinase (D3), a selenoprotein that plays an important role in thyroid hormone metabolism. Thyroid hormones influence a wide variety of biological processes in vertebrates. Their importance is most evident during prenatal and early neonatal development  QTL. The most plausible explanation is that DIO3 could play a role in the regulation of female fertility and/or on the survival of fertilized oocytes and embryos.Limited studies have described the effect of imprinted genes on litter size. An imprinted effect on litter size has been observed in mouse for the (predominantly) maternally expressed gene GRB10 The genotypic effects for the imprinted QTL suggest maternal expression  as predicted by the parental-offspring conflict hypothesis for genomic imprinting The higher than expected frequencies of the BA genotype of SNP marker ASGA0037226 in both populations was of interest because this genotype class was also favorable in terms of the traits studied in both populations (sows with a BA genotype had more offspring than sows with a AB genotype ). The reRecent publications reported an effect of the paternally expressed IGF2 gene on sow prolificacy traits In this study, we only considered imprinted genes which have been experimentally confirmed in human, mouse or other mammalian species. These more than 100 imprinted genes are located in 40 regions on the human genome . Fifteen of these regions were selected for genotyping see . The regThe location of the regions in the pig genome, orthologous to the imprinted regions in human plus 0.25 Mb at the 5′ and 3′ flanking sequence, were found by megaBLAST searches A 384-plex Golden gate SNP assay was developed to cover the fifteen selected regions. Twenty to 38 SNPs were allocated to each region. The number of SNPs allocated to the different regions depended on the number of imprinted genes in each region, on the size of the region and on the expected importance of the imprinted genes in the region on reproduction.  and Topigs (further denoted as population C2) were genotyped and their data were analyzed with the objective to detect genomic imprinting affecting reproduction traits. These populations were chosen because they had detailed information on fertility traits and because they were sufficiently large to allow for optimization of the study design.To enable accurate inference of allele origin, which involves inference of haplotypes, a sow was only selected when her father and more than two of her paternal halfsibs were available for genotyping. Available ancestors of a selected sow were also selected for genotyping.The pedigree of population C1 consisted of 6750 individuals, of which 4033 had phenotypes and in total 689 individuals from this population were genotyped. The pedigree of population C2 consisted of 10096 individuals, of which 3297 had phenotypes and in total 1050 individuals from this population were genotyped. On average, 4 generations of pedigree were available for the genotyped individuals of population C1 and 6 generations for the genotyped individuals of population C2.The phenotypes considered in this analysis were the total number of piglets born (TB), the number of piglets born alive (LB), the total weight of the piglets born in kilograms (TW) and the total weight of the piglets born alive in kilograms (LW). The weight traits TW and LW were expressed in kilograms and fewer observations were available for these traits than for the count traits TB and LB.The records of litters until the fourth parity of a sow were used in the analyses. A record of a specific trait was considered as outlier and excluded from the analyses when it deviated more than three standard deviations from the mean of that population. In population C1, 92 records for TB, 136 for LB, 10 for TW, and 8 for LW were considered as outliers. In population C2, 97 records for TB, 97 for LB, 43 for TW, and 35 for LW were considered as outliers. Outliers were removed because one outlier can have a dramatic effect on the p-values, in case outliers occur in genotype classes with only a few observations. On the other hand removing outliers might result in missing interesting findings. Therefore we compared for each company if genotype frequencies in the outliers and the data that was analyzed differed. This was not the case suggesting that outliers were randomly distributed across genotype classes. In addition, records for all four traits of a specific litter were excluded when TB or LB of that litter were 0. In population C1, no records were excluded for this reason. In population C2, the records of 712 litters were excluded for this reason.http://www.illumina.com). The Illumina's GenomeStudio 2009.1 framework Genotyping Module (v1.0) was used to score genotypes from the raw BeadXpress data. A manually refined genotype clustering file, based on 192 samples, was used for genotype scoring and the 384 SNPs were inspected to detect erroneous SNPs, which were excluded from further analyses. After excluding erroneous and monorphic SNPs, 309 SNPs remained for the association study.Samples from the two pig populations were supplied as hair or blood samples by the two breeding companies. DNA was isolated either from hair with the NucleoSpin tissue kits or from blood with the NucleoSpin blood kit, following the instructions of the manufacturers. The DNA concentration was determined with a NanoDrop Spectrophotometer and diluted or concentrated by evaporation to a working concentration of Mendelian inconsistencies in the genotype data were identified using the program Mendelsoft The parental origin of alleles were estimated using the program cvmhaplo y is a vector of phenotypic observations, X is the design matrix of the fixed effects, b is an unknown vector of fixed effects, Q is the design matrix of the effects of a specific marker which is explained below, q is an unknown vector of additive, dominance and imprinting effects of that marker. Matrix Z is the design matrix of the random additive genetic effects a and of the permanent environmental effects pe. A multivariate normal distribution with covariance matrix a, were A is the additive genetic relationship matrix calculated from the pedigree. A multivariate normal distribution with covariance matrix pe. Matrix M is the design matrix for the maternal effects, i.e. the mothers of the sows in our data. A multivariate normal distribution with covariance matrix v. A multivariate normal distribution with covariance matrix e.The univariate statistical analyses of the data were performed for each population and each trait separately. The following mixed effects model was fitted to the data using ASREML The fixed effects included in the model (apart from the marker effects) were a class effect accounting for the breed of the litter  (six levels in population C1 and 13 levels in population C2); a class effect accounting for parity of the sow (four levels in both populations); and a class effect accounting for the combination of farm, year and season (135 levels in population C1 and 333 levels in population C2).In an initial analysis, the model without the marker effects  to which each genotype belonged. In this notation, the first letter of the genotype indicates the allele inherited from the mother and the second letter the allele inherited from the father. Matrix S is a 4 by 3 contrast matrix of the additive, dominance and imprinting effect, as used by Hager et al. Design matrix S corresponds to the additive effect, the second column of S corresponds to the dominance effect and the third column of S corresponds to the imprinting effect. The four rows of S correspond to the four genotype classes.The first column of Q.Incremental F-ratios were calculated for the additive, dominance and imprinting effects of each marker, including the marker as the last fixed effect in the model. Following the decomposition of genetic variance by Fisher The significances of the marker effects where tested using the F-test statistic and the Kenward and Roger approximation for the denominator degrees of freedom as calculated by ASREML The q-values were calculated separately for each combination of population, trait, and genetic effect . The strength of evidence was expressed as the q-value of the test, following the notation of Storey and Tibshirani Supplemental File S1Infomation of the markers and P-values for each marker. The list of markers shows the markers included in the analysis, with their position on the reference genome build 9, the region in which they were located and other information. The list of P-values of the markers shows the P-value for the Additive (A), Dominance (D) and Imprinting (I) effect of each marker in each analysis (four traits x two breeding companies).(XLS)Click here for additional data file.


Modern genetics has been transformed by high-throughput sequencing. New experimental designs in model organisms involve analyzing many individuals, pooled and sequenced in groups for increased efficiency. However, the uncertainty from pooling and the challenge of noisy sequencing data demand advanced computational methods.ULTIPOOL, a computational method for genetic mapping in model organism crosses that are analyzed by pooled genotyping. Unlike other methods for the analysis of pooled sequence data, we simultaneously consider information from all linked chromosomal markers when estimating the location of a causal variant. Our use of informative sequencing reads is formulated as a discrete dynamic Bayesian network, which we extend with a continuous approximation that allows for rapid inference without a dependence on the pool size. MULTIPOOL generalizes to include biological replicates and case-only or case-control designs for binary and quantitative traits.We present MULTIPOOL is freely available at http://cgs.csail.mit.edu/multipool/.Our increased information sharing and principled inclusion of relevant error sources improve resolution and accuracy when compared to existing methods, localizing associations to single genes in several cases. M ULTIPOOL, a multi-locus method for analyzing high-throughput DNA sequencing reads obtained from large pools of phenotypically-extreme individuals.Advances in high-throughput DNA sequencing have created new avenues of attack for classical genetics problems. A robust method for determining the genetic elements that underlie a phenotype is to gather and group individuals of different phenotypes, interrogate the genome sequences of each group, and identify elements that are present in different proportions between the groups. We describe MULTIPOOL.We focus on model organism experiments where two strains are crossed and the progeny are grouped and pooled according to phenotype. We describe and model experiments for haploid organisms that are hybrids between two strains, but we note that the models we develop should generalize to more sophisticated crosses or diploid organisms. When two strains vary in a phenotype, analyzing progeny with extreme phenotypes should elucidate the genetic basis of the trait. The main idea is that polymorphic loci that do not affect the phenotype will segregate with approximately equal frequency in the progeny (regardless of phenotype), while loci that influence the trait will be enriched in opposite directions in the extreme individuals, according to the effect size of each locus. This approach assumes that the causal loci have sufficiently strong main effects to be detectable via any type of pooled analysis. This pooled study design is also referred to as "bulk segregant analysis"  in modelArabidopsis [Bulk segregant analysis with high-throughput sequencing has been applied in yeast to study drug resistance in , high tebidopsis  model sybidopsis .Pools may be selected from a single phenotypic extreme, opposite extremes, or one extreme and a control sample. Pools may also be obtained by grouping based on binary traits rather than quantitative phenotype extremes. Early studies used microarrays for pooled genotyping -10, but Pooled genetic mapping studies using high-throughput sequencing present a number of unique difficulties. The core statistical quantity of interest, the allele frequency in each pool, is observed only indirectly. The strain-specific read counts that are used to estimate the allele frequencies are corrupted by sampling noise at most reasonable sequencing depths, read mapping errors , referenULTIPOOL.However, the unbiased nature of genotyping via high-throughput sequencing results in nearly saturated marker coverage where almost all polymorphisms are queried. This avoids the laborious process of marker discovery and assay design required by earlier genotyping technologies. The dense marker coverage also allows for a high degree of information sharing, which motivates the methods underlying MPrevious statistical approaches to analyzing pooled genotyping data have focused on alternate regimes where genetic markers are relatively sparse and measurements are relatively accurate. Often, only single loci are tested for association, necessarily ignoring data from nearby markers. Additionally, single-locus methods encounter difficulties with missing data, such as regions that are difficult to sequence or map or have very few polymorphisms.p-values from local tests in [Earlier work applied hidden Markov models (HMMs) to fine mapping within small regions with fewer number of markers ,16, and tests in , while atests in . HoweverULTIPOOL is designed for experimental crosses and dense noisy genotyping, as obtained by sequencing, and handles datasets with tens or hundreds of thousands of markers. We develop a statistical model that can combine information across many nearby markers while accounting for the nonuniform noise levels introduced by varying sequencing depth and marker spacing. The specific advances we present with MULTIPOOL include:M• A model-based framework that allows for information sharing across genomic loci and incorporation of experiment-specific noise sources. These methods improve on previous approaches that rely on heuristic techniques to select sliding window sizes, which may sacrifice resolution.• Statistical tests using an information-sharing dynamic Bayesian network (DBN) that report robust location estimates and confidence intervals. The multi-locus methods allow for principled inference even in regions without strain-specific markers and reduce experimental noise when many markers are available.• Extensions of our method to any number of replicates and multiple experimental designs, within the same principled statistical framework.We develop inference methods for the pool allele frequency at a particular genome position, given the pooled read samples. First, we propose generative models which describe the experimental process. Next, these models are used to construct likelihood-based statistics to assess the significance of associations in multiple experimental designs.bwa[samtools[All sequencing reads from a particular pooling experiment are aligned to one strain's reference genome using the short read aligner bwa. To incr[samtools. A genom[samtools, and the[samtools.ULTIPOOL uses a probabilistic model that considers one chromosome at a time and explicitly models the effect that recombination and pool size have on neighboring allele frequencies. The model is a dynamic Bayesian network that describes the changing allele frequencies in the pool along a chromosome. The chromosome of interest is segmented into discrete blocks of equal size. A hidden state corresponding to each block reflects the pool allele frequency in the pool at that locus, varying along the genome as recombination causes random fluctuations. Each locus may emit sequencing reads according to its local pool allele frequency (hidden state). These reads may originate from multiple markers falling within the same region or a single marker. When there are no polymorphisms or mappable reads available in a region, the locus has no emissions and therefore the observed data do not directly constrain the hidden state at that locus. Finally, a particular locus may include the causal gene and therefore be directly associated with the phenotype. We assume there is only one causal locus in the analyzed region. For the genetic mapping problem, the causal locus is unknown and the key inference task is identifying its location and degree of association with the phenotype.MN individuals. An unknown causal locus is linked to the phenotype and displays association with allele frequency i by xi, i = {1, .., L}. The observed allele frequency measurements yi are obtained from the mapped sequencing reads. We also define di, the total informative reads at each locus. This quantity is determined by the local sequencing depth and number of mappable polymorphisms. The recombination frequency r gives the probability of an odd number of crossovers between adjacent genome segments in one individual in the pool. We do not model crossover interference, and therefore assume that recombination events are independent along the genome. The dependencies encoded in this model can be expressed as a graphical model, shown in Figure p. The population allele frequency is the allele frequency of the causal locus that would be observed in an infinitely-large pool (the population), and depends on the strength of the locus's association. Subsequent sections develop efficient methods for calculating likelihoods for all relevant model structures by reusing intermediate computations.The pool is composed of di can be calculated using the binomial distribution:The probability of observing a set of sequencing reads conditioned on the pool fraction at the locus and a total informative read count This formulation models the read count proportion exactly with a discrete model. An approximation, applicable to high read counts, can be obtained with a Gaussian distribution:Technical pooling variance that increases the local measurement noise, such as allele-specific PCR amplification bias, could be assumed to act in locus-independent manner and be modeled with increased variance in this expression.r is effectively the probability of a single recombination event occurring. We can determine the transition probabilities from xi to xi+1 by considering the k individuals that switch from the first strain to the second and the j individuals of the reverse case. We know k ~ Bin since each of the Nxi individuals with the first strain's ancestry at locus i will switch strain type when a recombination event occurs, with probability r. Similarly, j ~ Bin(N(1 - xi),r). Thus:In practice, the genome segments are chosen to be small enough so that N, we obtain an approximation for the transition probabilities:Employing normal approximations for the binomial distributions and dividing by This formulation shows that the latent allele frequencies form a first-order autoregressive Gaussian process with mean p:The causal locus node induces a particular distribution over hidden states, depending on the selected population allele frequency The normal approximation is:Inference of the hidden state values can proceed outwards from the causal locus, using the conditional independence structure of the model. We describe the algorithms in terms of standard HMM techniques, but note that a more general treatment in terms of message passing is also possible.xc (model structure) and population allele frequency p, is obtained by conditioning on the values of the causal locus:The observed data likelihood xc use the same graphical structure over the latent states xc: two chains with all rightwards arrows, separated by the conditioned node xc. Using this fact, we can compute the desired likelihoods with intermediate computations from a single graphical model.The first term in the sum operates on an HMM with rightwards arrows in its graph, while the second term operates on an HMM with leftwards arrows . This dominates the cost for the final step of computing p, which is O(NL). The quadratic dependence on the pool size renders the exact modeling of large pools prohibitive, motivating the continuous approximation in the next section.Running the forward-backward algorithm requires considering all transitions in each chromosome block, leading to a runtime quadratic in the size of the pool: The previous inference procedure applied to discrete hidden states where the pool composition is modeled exactly, but yielded inference algorithms that require time quadratic in the size of the pool. For large pools, we can relax this requirement and avoid the quadratic burden by modeling the allele frequency as a continuous value. The graphical model is linear-Gaussian since the transitions and observations are linear functions of the latent variables, subject to Gaussian noise. In a linear dynamical systems formulation, the model is:Where:vi can be approximated with the sample variance from the observed yi, depending on yi and di, or upper bounded by The per-locus observation noise for HMMs -23. The Where:The recursions begin with the stationary distribution parameters:The Kalman smoothing equations use the filtered results (forward estimates) to create estimates of the hidden state using the entire observation sequence, recursing backwards:Where:As in the discrete section, the posterior distributions of the latent states under a null model can be used to compute the desired data likelihoods for all possible causal models. Required integrals are computed numerically using a fixed number of points. Specifically:c requires O(L) time. This removes the dependence on the size of the pool that was present in the discrete method, allowing MultiPool to perform accurate inference in very large pools.Since the probability distributions during inference are represented with a constant number of parameters instead of a full vector (as in the discrete case), inference is more efficient. Specifically, computing the required quantities p. For each locus, we construct a likelihood ratio statistic comparing the hypotheses of association and no association:With these computations in place, we can compare all values of the causal locus and the trait association, measured by The simplification occurs because the likelihood under the noncausal hypothesis at any locus is the same, namely p' numerically and calculate the likelihood ratio for all positions of the causal locus by reweighting the posterior probabilities.We perform the maximization over p:We can analyze replicate experiments by forming a coupled dynamic Bayesian network. This analysis present two replicates, but the methods generalize to any number of coupled experiments. In this situation, the same sampling distribution is induced at the shared causal locus in two coupled chains. The joint data likelihood factors since the chains are conditionally independent given the selection node p' must consider the product of the data likelihoods in the replicates. For designs where paired experiments are expected to show opposite effects, each experiment selects an optimal population allele frequency p. In this case, the null hypothesis is the coupled model where the two experiments share the same population allele frequency. The likelihood ratio is:The maximization over The numerator is the product of two single-experiment maximizations, while the denominator is the coupled model likelihood that was presented for replicate analysis.MultiPool reports log10 likelihood ratios (LOD scores in the genetics community), maximum-likelihood estimates (MLE) of the causal locus location, and approximate credible intervals for the location of the causal locus. Assuming a uniform prior over causal locus locations, p at its MLE, but could alternately integrate it out. Therefore, we can compute multi-locus statistics that include information from the entire dataset in experiments where multiple pools are available.Using these results, MultiPool versus single-locus tests on deeply-sequenced pools, we conducted a series of simulations. A causal locus was chosen with population allele frequency p = 0.75 and many pools of sizes N = 100, 1000, and 10000 were created. SNP locations and relative per-SNP sequencing depths were calculated from experimental datasets in yeast. Average read coverage (sequencing depth) was varied from 10X to 150X, and 100 datasets of each type were simulated. The MLE causal allele location was calculated with the single-pool DBN model. The single-locus test analyzed allele frequencies computed with 1kb sliding windows, based on the method in [N = 100 cases, the mapping accuracy is predominantly controlled by the small pool size. This leads to little improvement with increased sequencing depth. The larger pools show higher accuracy with increased sequencing depth, but MultiPool is always more accurate with a lower sequencing requirement. These simulations were conducted without additional read mapping noise or other noise sources, and so the absolute results should be interpreted conservatively.In order to understand the benefit of ethod in . The rooWe also analyze pooled sequencing data recently generated by two groups ,3. The gMultiPool to a likelihood ratio computed using allele frequencies calculated by summing allele read counts in sliding windows. The data likelihoods under the causal and noncausal models are calculated according to the model in [In cases where the associated region is localized to a single gene, we compare the LOD scores from model in , with thmodel in .RAD5 and MKT1 were validated as affecting 4-NQO resistance with follow-up experiments, so we use them as test cases for our model. The control pools showed no association around the validated loci, so we applied MultiPool's one-pool test for association using the continuous model. Table MultiPool and sliding-window tests.The first set of large pools was used to characterize the genetic basis of resistance to the DNA-damaging agent 4-NQO. The genes MultiPool localizes RAD5 to within the gene body, without a dependence on choosing an appropriate sliding window size. The 90% credible interval of the location contains six genes, centered on RAD5. A localization example using one replicate is shown in Figure MKT1 is localized to within 3 kb of the gene body, with the 90% credible interval covering MKT1 and eight other genes.r in our model). In this study, the genes IRA1 and IRA2 were verified as affecting heat sensitivity with direct assays. Table MultiPool and sliding window methods. IRA2 is localized to within the gene body, but the predictions for IRA1 are consistently upstream of the gene's location. Upon further investigation, the peak around IRA1 appears to contain another (untested) associated locus. Figure IRA2 in all datasets, but do not include IRA1. This may support the hypothesis of another linked gene in the associated region.The second set of large pools was constructed to study the genetics of heat tolerance, using repeated crosses to reduce linkage disequilibrium . Other DBN: dynamic Bayesian network; HMM: hidden Markov model; LOD: base 10 logarithm of odds; MLE: maximum likelihood estimate; QTL: quantitative trait locus; RMSE: root mean square error.The authors declare that they have no competing interests.MDE and DKG conceived and designed the research. MDE performed the research. MDE and DKG wrote the paper.


In genetic association study, especially in GWAS, gene- or region-based methods have been more popular to detect the association between multiple SNPs and diseases (or traits). Kernel principal component analysis combined with logistic regression test (KPCA-LRT) has been successfully used in classifying gene expression data. Nevertheless, the purpose of association study is to detect the correlation between genetic variations and disease rather than to classify the sample, and the genomic data is categorical rather than numerical. Recently, although the kernel-based logistic regression model in association study has been proposed by projecting the nonlinear original SNPs data into a linear feature space, it is still impacted by multicolinearity between the projections, which may lead to loss of power. We, therefore, proposed a KPCA-LRT model to avoid the multicolinearity.Simulation results showed that KPCA-LRT was always more powerful than principal component analysis combined with logistic regression test (PCA-LRT) at different sample sizes, different significant levels and different relative risks, especially at the genewide level (1E-5) and lower relative risks . Application to the four gene regions of rheumatoid arthritis (RA) data from Genetic Analysis Workshop16 (GAW16) indicated that KPCA-LRT had better performance than single-locus test and PCA-LRT.KPCA-LRT is a valid and powerful gene- or region-based method for the analysis of GWAS data set, especially under lower relative risks and lower significant levels. It is commonly believed that genetic factors play an important role in the etiology of common diseases and traits. With rapid improvements in high-throughout genotyping techniques and the growing number of available markers, genome-wide association studies (GWAS) have been promising approaches for identifying common genetic variants. The first successful wave of GWAS has reproducibly identified hundreds of associations of common genetic variants with more than 100 diseases and traits, including age-related macular degenerative diseases  and typeT2 test [However, given the SNPs allocated into genes or regions, the issue of how to evaluate genetic association for each candidate gene or genome region remains. To examine whether multiple SNPs in the candidate gene or region are associated with disease or trait, several multi-marker analysis methods have been developed, including haplotype-based methods ,17, HoteT2 test ,19, prinT2 test -23, and T2 test ,24,25. EHowever, one cannot assert that linear PCA will always detect all structure in a given genomic data set. If the genomic data contains nonlinear structure, PCA will not be able to detect it . FurtherKPCA has been studied intensively in the last several years in the field of machine learning, face recognition and data classification, and has been claimed success in many applications -30. EspeM SNPs in a candidate gene or specific genome region of interests have coded values {xi ∈ RM | i = 1,2,...,N}, where N represents sample size giving a genetic model (assuming additive model here). PCA diagonalizes the covariance matrix of the centered observations xi, As a traditional multivariable statistical technique, PCA has been widely applied in genetic analysis, both for reduction of redundant information and interpretation of multiple SNPs. The basic idea of PCA is to efficiently represent the data by decomposing a data space into a linear combination of a small collection of bases consisting of orthogonal axes that maximally decorrelate the data. Assuming that To do this, one has to solve the following eigenvalue problem:ν are the eigenvectors of C, and λ are the corresponding eigenvalues. As ν must lie in the span of {xi ∈ RM | i = 1,2,...,N}, hence (2) is equivalent towhere a =  and b =  is defined as where the dot product of two vectors F byGiven the observations, we first map the data nonlinearly into a feature space x1),...,Φ(xN), is centered, i.e. Again, we make the assumption that our data mapped into feature space, Φ,...,Φ(xN). This implies that we may consider the equivalent equationBy the same argument as above, the solutions ai  such thatand that there exist coefficients Substituting (3) and (5) into (4), we arrive atα denotes the column vector with entries α1, ..., αN, and K is a symmetric N × N matrix defined bywhere It has a set of eigenvectors which spans the whole space, thusα of equation (6).gives all solutions λ1 ≤ λ2 ≤ ... ≤ λN represent the eigenvalues for the matrix K with α1, α2, ..., αN being the corresponding complete set of eigenvectors. λp is the first nonzero eigenvalue. We do the normalization for the solutions αp, ..., αN by requiring that the corresponding vectors in F be normalized, i.e. νk · νk = 1 for all k = p, p + 1, ..., N. Based on (5), (6) and (8), this translates intoAssume k in F to do principal component extraction. Suppose x is the SNP set within previously defined gene or genome region of an individual, with an image Φ(x) in F, thenWe need to compute projections on the eigenvectors νare its nonlinear principal components corresponding to Φ.xi) in explicit form - they are only needed in dot products. We, therefore, are able to use kernel functions for computing these dot products without actually performing the map Φ: for some choices of a kernel k, by methods of functional analysis, it can be shown that there exists a map Φ into some dot product space F (possibly of infinite dimension) such that k can compute the dot product in F. This property is often called "kernel trick" in the literature.Note that neither (7) nor (10) requires Φ principal component scores of the SNPs, respectively. The value of L can be chosen such that the cumulative contributing proportion of the total variability explained by the first L PCs (λ1 + λ2 + ···+ λL)/(λ1 + λ2 + ··· + λM) exceeds some threshold. For comparison, we set the same threshold of 80% in both PCA-LRT and KPCA-LRT as Gauderman et al [where an et al .H0) and alternative hypothesis (H1). The corresponding steps for the simulation are as follows:To assess the performance of KPCA-LRT and compare it with PCA-LRT, we apply a statistical simulation based on HapMap data under the null hypothesis (Step 1. Download the phased haplotype data of a genome region from the HapMap web site (http://snp.cshl.org): we select the Protein tyrosine phosphatase, non-receptor type 22 (PTPN22) gene region to generate the simulating genotype data of CEU population using HapMap Phase 1& 2 full dataset. This region is located at Chr 1: 114168639..114197803, including 11 SNPs. Figure 2 structure and minor allele frequencies (MAF).Step 2. Based on the HapMap phased haplotype data, we generate large samples with 100 000 cases and 100 000 controls as CEU populations using the software HAPGEN [H0, we set the relative risk per allele as 1.0 to assess the type I error. Under H1, different levels of relative risks are set  to assess the power. The SNPs in this region are coded according to the additive genetic model.e HAPGEN . To inveStep 3. From the remained SNPs, we sample the simulation data and perform the PCA-LRT and KPCA-LRT under different sample sizes N  using the R packages kernlab (http://cran.r-project.org/web/packages/kernlab/index.html) and Design (http://cran.r-project.org/web/packages/Design/index.html). Under H0, we repeat 10 000 simulations at two significant levels (0.05 and 0.01). Under H1, for each model with a given relative risk, we repeat 10 000 simulations at four significant levels .The proposed method is applied to rheumatoid arthritis (RA) data from GAW16 Problem 1. The data consists of 2062 Illumina 550 k SNP chips from 868 RA patients and 1194 normal controls collected by the North American Rheumatoid Arthritis Consortium (NARAC) . At presTo illustrate the performance of PCA-LRT and KPCA-LRT, we mainly focus on four special regions in chromosome 1, within the genes PTPN22, ANKRD35, DUSP23, RNF186 involved, respectively. The reasons are as follows: 1) Both the PTPN22 gene  and ANKRD35 gene have been reported to be associated with RA -43; 2) DH0 are shown in Table α = 0.01, α = 0.05) under different sample sizes. This suggests that both the models perform well under null hypothesis.Simulation results under th SNP (rs1746853) as the causal variant, Figure When defining the 6With the same causal variant as above, Figure These simulation results indicate that the powers of KPCA-LRT are always higher than PCA-LRT at given significant levels, sample sizes and relative risks. Particularly, under lower relative risk (1.2 and 1.3) and smaller significant levels (1E-5 and 1E-7), KPCA-LRT is more powerful than PCA-LRT.Table In genetic association study, especially in GWAS, in order to avoid the collinearity among SNPs and reduce the false positive rate caused by multiple testing, several groups have proposed PCA-based methods and found that these methods are typically as or more powerful than both single locus test and haplotype-based test -23. HoweTo compare the three methods , the four regions from the RA data in GAW16 Problem 1 Table  are consThe four genes involved in the regions for real data analysis are selected based on prior researches and Gene Ontology . The defThere are several limitations about the proposed method. First, only one causal SNP is considered in present work. Second, how to fix the kernel function with appropriate parameters for each data is still a theoretical problem. Third, when the effect size is smaller . Results from both simulation studies and application to real data show that KPCA-LRT with appropriate parameters is always as or more powerful than PCA-LRT, especially under lower relative risks and significant levels.The authors declare that they have no competing interests.QSG, YGH, ZSY, JHZ, BBZ and FZX conceptualized the study, acquired and analyzed the data and prepared for the manuscript. All authors approved the final manuscript.