The heritability of a trait is the proportion of its variance explained by genetic factors; it has historically been estimated using familial data. However, new methods have appeared for estimating heritabilities using genomewide data from unrelated individuals. A drawback of this strategy is that population stratification can bias the estimates. Indeed, an environmental factor associated with the phenotype may differ among population subgroups. This factor being associated both with the phenotype and the genetic variation in the population would be a confounder. A common solution consists in adjusting on the first Principal Components (PCs) of the genomic data. We study this procedure on simulated data and on 6000 individuals from the Three-City Study. We analyse the geographical coordinates of the birth cities, which are not genetically determined, but the heritability of which should be overestimated due to population stratification. We also analyse various anthropometric traits. The procedure fails to correct the bias in geographical coordinates heritability estimates. The heritability estimates of the anthropometric traits are affected by the inclusion of the first PC, but not by the following PCs, contrarily to geographical coordinates. We recommend to be cautious with heritability estimates obtained from a large population. The heritability of a quantitative phenotype is the proportion of its variance explained by genetic factors. The concept of heritability can be traced back to the pioneering works of Galton in the nineteenth centuryHeritability estimates have long been based on family data4The research of the genetic polymorphisms causing the variability of the phenotypes with a strong genetic component was carried notably through numerous Genome-Wide Association Studies (GWAS). Hundreds of Single Nucleotide Polymorphisms (SNPs) have been found associated with complex traits689101112The missing heritability problem triggered the development of methods using genome-wide data to obtain heritability estimates from unrelated individuals1317181920The common usage is to include 10 or 20 PCs in the model with fixed effects. There are however no theoretical grounds on which this number of PCs is chosen; more PCs could be included without damaging the estimates. In this paper, we explore the variability of genomic heritability estimates with the number of PCs included as fixed effects in the model, and the ability of this method to effectively correct for population stratification.We use simulated data to assess the properties of the method, which we also apply on the Three-City (3C) dataWe also analyse anthropometric phenotypes: height which is the historical example of a highly heritable trait225262829303132h2 of several traits. The decompositions of phenotypic variance are presented in graphics with number of Principal Components included with fixed effects in the model as abscissae and the proportion of explained variance as ordinates. The estimated proportion of variance explained by the fixed effects is displayed in dark gray. The light gray and white colors are respectively the estimated proportions of genetic and residual variances.Using linear mixed models, we estimate variance components and heritabilities H0: h2 = 0.For the 3C phenotypes, we also give Tables with numerical value of the parameter estimates, their standard errors, and the likelihood ratio test for p of PCs included in the model with fixed effects varying from 0 to 2000.Phenotypes were simulated under the linear mixed model, with a non-zero effect on the first 10 PCs (see details in Methods section). They were analysed with a number Variance estimations have also been made on several quantitative traits from the 3C study. First of all, we considered as quantitative phenotypes the longitude and the latitude of the birth cities. Geographical coordinates of all birth cities are displayed on the map of France in the first panel of The mean and the standard deviation of each analyzed variable are given in p = 2000 PCs in the model with fixed effects. When no PCs are included in the model, it is estimated that 100% of the variance is genetic both for longitude and latitude. When adding PCs to the model, this estimated proportion decreases slowly for the longitude  with the first few PCs included in the model; it then decreases slowly, as observed for latitude: it is still close to 36% for p = 50 PCs, decreasing to 16% for p = 2000. For both geographical coordinates, the first PC explains a non-negligible proportion of variance (29% and 31% for longitude and latitude respectively). Once the first PC is included, the proportion of variance explained by the PCs increases slowly with p. This observation is consistent with the correlation values of the geographical coordinates with the first few PCs  is significant for the heritability of both longitude and latitude , the genetic variance is 22%, 13% and 12% of the total variance; the inclusion of a single PC in the model (p = 1) decreases these values to 19%, 8% and 8% respectively. Including up to p = 50 PCs does not modify much these proportions. For p up to 2000, the variance proportion estimates fluctuate around the previously cited values.In The estimated proportions of variance for weight and BMI are displayed in p, the estimates of σ2, τ, and h2, together with their standard error, and the likelihood ratio test for the null hypothesis H0: h2 = 0.p = 20) almost does not change the estimated values nor their standard errors. The likelihood ratio tests for h2 = 0 are significant.Without stratification correction, height heritability is estimated to 46% with standard error of 6.1%; as soon as one PC is included in the model, this value drops to 39% (standard error 6.6%). Inclusion of more PCs  and 19% (standard error 6%).p = 1 PC in the model, it is only 10%, which drops to 9% for p = 2 (with a standard error of 6.3% or 6.4%). The likelihood ratio test is no longer significant.Head circumference heritability is estimated to 17% (standard error 5.7%) without population stratification correction. With inclusion of Waist-to-hip ratio heritability is estimated to 20%  with the inclusion of one PC in the model. The likelihood ratio tests stay significant.p = 100 or 500 is not harmful for the precision of the estimates. There is no practical reason to limit to small values of p.Our analyses of simulated data with up to 2000 PCs included as fixed effect show that the heritability estimates precision are not much impacted when an important number of PCs are included in the model with fixed effects. Of course, the size of the sample matters, however as soon as a few thousands individuals are included in the analysis, taking p.We were surprised to obtain a genomic heritability estimate of 100% for latitude and longitude: as they are correlated to the first few PCs, we were expecting a positive heritability estimate, but not that large. We were also expecting the estimated heritability to vanish  after inclusion of a few PCs in the model. Instead of that, if the latitude heritability drops to 68% after inclusion of the first two PCs, adding more PCs only results in a slow decrease, and the heritability remains highly significant. The longitude behaves in a worse manner, as there is no initial drop as observed for the longitude. In both cases, the slow decrease of heritability is accompanied by a slow increase of the proportion of total variance due to the PCs included with fixed effects. The Likelihood Ratio Test (LRT) shows that the heritability is significantly positive for all values of et al.et al.Browning Heritability estimates obtained for anthropometric traits are globally compatible with the results from the literature. It is interesting to note that height and waist-to-hip ratio seem sensitive to population stratification, even if marginally. Head circumference is more affected, as its estimated heritability drops from 17% to 9% with the inclusion of the first two PCs, and the LRT is no longer significant. All these traits are weakly but significantly correlated with the geographical coordinates of the birth cities cf , but so p = 2: this falls in line with our previous findings on the fact that the inclusion of the first PCs may not be sufficient to fully correct for population stratification. One should consider seriously to include the geographical coordinates of the place of birth with fixed effect in the model, when they are available.We performed additional analyses were we included the geographical coordinates as covariates when analysing height, head circumference, and waist-to-hip ratio : the droEpidemiologists use to take into account possible center effects by including indicator variables for the centers. We performed an additional analysis of height, head circumference and waist-to-hip ratio with fixed effects for the centers . As obseWe made the choice of using LD-pruned data to compute the PCs included for population stratification. We could have used the whole data to this aim; this does not alter significantly our conclusions. The results of such analyses are displayed in the A procedure has been proposed to detect the presence of cryptic relatedness and population stratificationA recent work argued that considering the genotype matrix as fixed in the linear mixed model, while in reality the genotypes are sampled from the population, is a cause of unreliability of genomic heritability estimates, and that this is aggravated by the presence of population stratificationWe consider the linear mixed modelwith τ and σ2 are estimated by Restricted Maximum Likelihood (REML)394041The variance components X. This is tantamount to defining the heritability as the proportion of genetic variance in the phenotype variance yet unexplained by known covariates, as in ref. Y in three parts: the variance explained by the covariates in X, the genetic variance τ, and the remaining variance σ2.The heritability is usually estimated by h2 and this decomposition of the variance in three components.Estimating the variance due to the covariates by the empirical variance of the components of p principal components of the kinship matrix estimated only on a submatrix Z1 of Z, obtained by pruning SNPs from Z, to retain approximately 50000 SNPs in low linkage disequilibriumTo correct for population stratification, we include the first PCi the ith principal component vector (n × 1) and βi the fixed effect of the ith PC.with p varying from 0 to 2000 (for n = 5793).We will use values of http://www.three-city-study.com/genetic-studies.php).The 3C study−8 were removed.Quality Control was performed using PLINKAfter Quality Control, there are 5793 individuals  and 509931 autosomal SNPs left.The Principal Components for population stratification correction are computed on LD-pruned data, where we kept only SNPs with a minor allele frequency higher than 5%, and in mutual LD inferior to 0.1. We also removed SNPs in the long-range LD regions defined in ref. Two types of traits are available; the longitude and the latitude of the birth cities which were retrieved from their zipcode . The remaining variance is equally distributed between genetic and environmental effects .We simulated phenotypes according to the model (1), using the 3C genomic data: How to cite this article: Dandine-Roulland, C. et al. Accuracy of heritability estimations in presence of hidden population stratification. Sci. Rep.6, 26471; doi: 10.1038/srep26471 (2016).


Genome-wide SNPs collectively account for, on average, about half of twin heritability across cortical regions (N=466 twins). We find enriched genetic effects in or near genes. We also observe that SNPs in evolutionarily more conserved regions contributed significantly to the heritability of cortical surface area, particularly, for medial and temporal cortical regions. SNPs in less conserved regions contributed more to occipital and dorsolateral prefrontal cortices.Little is known about how genetic variation contributes to neuroanatomical variability, and whether particular genomic regions comprising genes or evolutionarily conserved elements are enriched for effects that influence brain morphology. Here, we examine brain imaging and single-nucleotide polymorphisms (SNPs) data from ∼2,700 individuals. We show that a substantial proportion of variation in cortical surface area is explained by additive effects of SNPs dispersed throughout the genome, with a larger heritable effect for visual and auditory sensory and insular cortices ( How genetic variation contributes to brain morphology is still poorly understood. Here Chen et al. combine brain imaging with single-nucleotide polymorphism data to discover that a substantial degree of cortical variation is derived from underlying genetic differences. To understand the complexity of human higher cognition, it is essential to study the properties of the cerebral cortex13451One fundamental question is the extent to which neuroanatomical variability among individuals is caused by genetic differences. Twin and pedigree designs have been used to estimate the ‘heritability' of a phenotype by examining the resemblance of the phenotype between relativesh2)12Technological advances now allow assay of individuals for millions of single-nucleotide polymorphisms (SNPs) spanning the whole genome99To investigate the polygenic contribution of common SNPs to cortical structures, we apply the GCTA method to a combined sample from five cohorts. Raw imaging and genotype data from all study cohorts are processed with a standardized protocol to minimize data heterogeneity. The analysis for estimating SNP heritability is potentially sensitive to population structure . It is therefore typical to restrict the analysis to unrelated individuals of a single genetic ancestryPhenotype definition is critical for all genetic association studies, especially in brain imaging genetic studies, due to the high dimensionality of cortical measures (∼0.3 million points per subject)6In our five-cohort sample, we found that a substantial proportion of variation in surface area in almost all of the 12 regions is captured by all autosomal SNPs after accounting for global cortical size. A few cortical regions, such as the insula, visual and auditory sensory regions, including superior temporal and occipital cortices, have a high SNP heritability of up to ∼0.45 (s.e. 0.12) . We repoWe next sought to compare the SNP heritability with twin heritability estimated from 466 twins. Heritabilities of the same cortical imaging phenotypes were estimated in a classical twin model implemented in the OpenMx software suite. The results of twin heritability estimates (additive genetic variances) across different cortical regions are shown in We partitioned the variance explained by all the SNPs into genic and intergenic regions across all autosomal chromosomes. We defined genic boundaries as 20 kb upstream and downstream from the 3′ and 5′ untranslated regions (UTRs) of each geneConservation scores were derived from multiple alignments of placental mammal genomes to the human genomesr=0.58 indicates that substantial variation can be attributed uniquely to each .We also found a highly significant correlation between the variance explained by genic SNPs and variance explained by more conserved SNPs , and betAlthough identifying genetic determinants of the human brain is an active area of research202225SNP heritability estimates quantify the overall contribution of the additive effects of all SNPs, which provides a lower bound of the narrow-sense heritability of the trait estimated in pedigree studies, since pedigree information captures the effects of all genetic variants on phenotypic similarity27Obtaining evidence for the polygenic architecture of complex traits provides a rationale for further dissecting the contribution of particular genomic regions to phenotypic expression1017The 1000-fold difference in cortical surface area between humans and mice may contribute to our complex behaviours1124We also found a highly significant correlation between the variance explained by genic SNPs and variance explained by more conserved SNPs , and betWe show that a substantial proportion of the heritable component of the cortex resides among common variants that can be interrogated via current genome-wide genotyping arrays. By leveraging genic and conservation annotations we were able to reveal that particular genomic regions are enriched for variants that influence variation in cortical surface area. Each cortical region appears to have elements of region-specific genetic architecture, which might relate to functional specialization of the cortical regions. Elucidating the sources of these genetic effects will allow investigators to prioritize resources for future investigations. Cortical surface area is similar to other complex traits in terms of polygenicity distributed among common variants and genetic effects enriched in genic regions. Yet, the human brain is a uniquely complex phenotype, in that its genomic properties appear as complex as its functional capacity. In this light, it should be acknowledged that beyond polymorphisms and structural variants in the genome, epigenomicsA total of 3,696 subjects with available and sufficient quality MRI scans from 5 cohorts were analysed. We removed non-European descents and related individuals. The combined sample of five cohorts is made of 605 subjects from the Thematically Organized Psychosis study 39The samples for the twin analysis was part of the Vietnam Era Twin Study of Aging (VETSA) studyEach study was approved by the local Institutional Review Board: South East Norway  and Mid Norway (HUNT) Regional Ethical Committee (HUNT), and UC San Diego (Paediatric Imaging Neurocognition and Genetics and VETSA).P<1 × 10−6), were excluded. Individual samples showing an over- or underabundance of heterozygosity (>5 s.d. from the mean) were labelled as poor quality and also excluded from subsequent analysesr2) was provided by Minimac and a threshold of r2>0.5 was used to declare successful imputation.All studies were genotyped using different commercial arrays. Standard genome-wide association quality control measures were applied to each study individually using the Plink toolset42http://freesurfer-software.org/). The cortical surface was reconstructed to measure surface areas at each surface location  using a semi-automated approach15MRI scans were analysed with software developed at the University of California, San Diego, Multi-Modal Imaging Laboratory based on the freely available FreeSurfer software package  based on the MRI scans of over 400 twins6To account for global effects, we divided the area measure of each location by the total surface area, so that the observed effects were specific to region of interest rather than having global associations with total surface area6We used a mixed linear model to fit a covariance structure of GRM to a vector of measured phenotypes via the GCTA tool9As described further in the n × n identity matrix, which assumes independence of environmental effects  and measurement error across individuals. Estimates where G is a matrix of kinship or genetic correlation coefficients and I is the h2), since it is based on the SNPs used to construct the GRM12The resulting estimate is referred to as ‘SNP' or ‘chip heritability' . This definition was used previouslyWe partitioned the variance explained by all of the SNPs into low- and high-conserved regions of the whole genome based on conservation annotation. We obtained a conservation annotation database from the UCSC Genome Browser hg19 assembly. The conservation scores were derived from alignments of placental mammals to human genome. PhastCons is a hidden Markov model-based method that estimates the probability that each nucleotide belongs to a conserved element, based on the multiple alignmentsWe assigned weights to conservation scores based on the LD information. We applied the pairwise LD matrix to the vector of phastCons scores. We expect that SNPs with the LD-weighted conservation annotation show more consistent and less noisy association signals. After the LD weighting, 48,523 of the ∼2.4 million SNPs had no scores and were eliminated from the subsequent analysis. We selected the median as a threshold to partition the genome evenly into low- and high-conserved SNPs (∼50%). We estimated the proportion of variance explained by low- and high-conserved genomic regions. The results are shown in How to cite this article: Chen, C.-H. et al. Large-scale genomics unveil polygenic architecture of human cortical surface area. Nat. Commun. 6:7549 doi: 10.1038/ncomms8549 (2015).Supplementary Figures 1-3, Supplementary Tables 1-9, Supplementary Methods and Supplementary References


To find genetic association between complex diseases and phenotypic traits, one important procedure is conducting a joint analysis. Multifactor dimensionality reduction (MDR) is an efficient method of examining the interactions between genes in genetic association studies. It commonly assumes a dichotomous classification of the binary phenotypes. Its usual approach to determining the genomic association is to construct a confusion matrix to estimate a classification error, where a binary risk status is determined and assigned to each genotypic multifactor class. While multi-class phenotypes are commonly observed, the current MDR approach does not handle these phenotypes appropriately because the thresholds for the risk statuses may not be clear. In this study, we suggest a new method for estimating gene-gene interactions for multi-class phenotypes. Our approach adopts the index of dissimilarity (IDS) as an evaluation measure. This is analytically equivalent to the common association measure of balanced accuracy (BA) for the binary traits, while it is not required to determine the risk status for the estimation. Moreover, it is easily expandable to the generalized index of dissimilarity (GIDS), which has an explicit form that can handle any number of categories. The performance of the proposed method was compared with those of other approaches via simulation studies in which fifteen genetic models were generated with three class outcomes. A consistently better performance was observed using the proposed method. The effect of a varying number of categories was examined. The proposed method was also illustrated using real genome-wide association studies (GWAS) data from the Korean Association Resource (KARE) project. Recently, genome-wide association studies (GWAS) have been popular for finding the association between a single nucleotide polymorphism (SNP) and complex traits. The traditional analysis tested a single SNP at a time and selected top few SNPs. However, this single-locus approach has evolved into a multiple-loci approach because most complex diseases are associated with multiple genes and their interactions –3.Several statistical methods can be used for detection of gene-gene interactions. The multifactor dimensionality reduction (MDR) method is a non-parametric approach that can efficiently identify higher-order interactions between genes ,5. OrigiMDR has been expanded to be applicable to various types of phenotypic traits. For quantitative traits, generalized MDR (GMDR) accomplishes this by covariate adjustment , while qSome phenotypes may consist of multiple categories which are not ordinal. For example, a combination of systolic and diastolic blood pressure is used as an indicator for hypertension, and the classification for the two combined is more complex than a simple ordinal in that its classification depends on ‘AND’ and ‘OR’ logics . HoweverOur aim in this paper is to propose a new MDR extension to estimate the genomic association of a gene-gene interaction with a multi-class phenotype. Our approach uses a measure that may be utilized for a multi-class phenotype without the need to explicitly construct a confusion matrix. We first introduce the index of dissimilarity (IDS) and the generalized IDS (GIDS) measures . We thenI.The index of dissimilarity (IDS) was originally conceived as a measure of segregation to quantify the extent of uneven distribution of two population groups in several areas of a city, such as census tract . It is dnij represents the number of subjects in the ith area and jth group, IDS can be interpreted as the proportion in one group or the other that must be moved in order to achieve an even distribution [Because ribution . A generJ = 2, the GIDS is reduced to the IDS as shown below.Note that the denominator in this equation defines the maximum possible value of the corresponding numerator, so that GIDS would have values between 0 and 1. When i and j should have an ith multi-locus genotype with a jth multi-class phenotype. Therefore, the numerator of the GIDS in Eq  utilizes the concept of risk status as depicted on the left side of Positive (P) or negative (N) classifications are determined by outcome classes corresponding to ‘case’ or ‘control’. As shown in Eq , risk stBA is then defined within the above schematics.The IDS may be applied to data with case and control trait groups without considering the risk status, as below.ncase = TP + FN and ncontrol = TN + FP, Eq (Using Eqs  and 5),,5), alon+ FP, Eq  can be rThe IDS and BA are now shown to be analytically equivalent except in the spanning range. Knowing that the possible range of BA is from 0.5 to 1, IDS spans the values from 0 to 1: absolutely no association corresponds to 0.5 for BA and 0 for IDS, while a maximum association gives rise to a value of 1 for both BA and the IDS.Estimation of genomic association for data with more than two outcome classes, as depicted on the right side of We conducted two simulation studies and analysed a dataset from a real genome-wide association study. For simulation I, a genetic dataset with a categorical trait was generated. For simulation II, a situation was considered in which the categorical trait was derived from the classification of the continuous variable. Data from the Korean Association Resource (KARE) project  was analvj, was fixed as 0.3, 0.4, and 0.3, throughout the models. With those prevalence and odds ratios, the set of penetrance, pj|i, should be determined. Then the penetrance was used as the probability of assigning the phenotypic class to each simulated sample. Minor allele frequencies (MAFs) were set as 0.5 for the first two patterns, and 0.3 for the rest of them. In all, 5×3 = 15 models were considered. For each model, 100 simulated data files were generated, with a sample size of 400, and 20 SNPs.A genetic dataset with a categorical trait of three classes was generated to demonstrate the ability of the GIDS to correctly identify the causal pair. A two-way interaction between single nucleotide polymorphisms (SNPs) was considered with a single causal pair intended. As depicted in 2 are compared in χ2 are essentially indistinguishable except with model 33, 42 and 43. Excluding them, overall hit ratios for the three kinds of locus classifiers were estimated to be 0.81, 0.96, and 0.94, respectively, for both GIDS and χ2. Here model 43 means the third variation within the pattern 4, plotted as the third one of the three grouped as model 4. Including them, the performance of χ2 was about 2% higher than that of GIDS. Only these three, out of all 15 models, turned out not to have cells with penetrance smaller than 10%. χ2 seems to take more advantage of the lack of extreme cell counts than GIDS. It can be concluded, with simulation I, that GIDS is able to estimate correct genomic association for multi-class phenotypes while showing the best performance for the order of the true causal interaction, which is 2-order with this simulation data.In Quantitative phenotype was generated in simulation II. Now the penetrance table dictated the choice of the distribution function according to Eq , yieldiny represents the phenotype value associated with two interacting SNPs, while fij and i and j have the values of 0, 1, 2 or AA, Aa, aa, making 9 distinct values for ij. With a single causal pair intended, penetrance table was taken from the models in [fij and L and σH in the distribution function were chosen independently out of 0.8, 1.0, and 1.2 such that 9 combinations could be formed. The parameter α was chosen as 0.4 so that the normal distributions could overlap adequately. There were total of 70 penetrance models [Here odels in . Accordie models , which wJ, is shown. The filled triangles, squares, and circles indicate the evolved GIDS with J = 2, 3, and 4, respectively, which means that the phenotypes are classified into 2, 3, and 4 categories. Effect of varying number of the trait classes, due to altered classification criteria, on the performance of the GIDS would be demonstrated by comparing the performances of GIDS-J2, -J3, and -J4. Because GIDS is intended for multi-class phenotypes that are intrinsically categorical or classified beforehand from a quantitative trait, it does not provide its own categorization scheme. While the actual classification rules or agreements will vary from phenotype to phenotype, we applied a simple ‘beforehand’ categorization rule such that the quantitative trait distribution would be divided into 2, 3, or 4 partitions with similar areas. Specifically we used the thresholds of (μ), , and  for the categorization of J = 2, 3, and 4, respectively. Here μ and σ represent the overall mean and standard deviation of the distribution where z = 0.43, 0.67 for J = 3, 4, respectively. A clear distinction can be seen between GIDS-J2 and GIDS-J3, or -J4. Because the dataset was prepared by mixing three Gaussian distributions, J = 3 or 4 may be expected to be more appropriately categorized than J = 2. A consistently better performance has been observed from GIDS-J3, and -J4 than from GIDS-J2. Recalling that the three datasets analyzed were made from a single original dataset, the performance difference between them may be interpreted as the correct reflection, by GIDS, of the adequacy of the categorization.In In χ2 tests is shown in χ2 also examines the deviation from the expected distribution similarly to GIDS. GIDS consistently showed better performance than χ2, regardless of the number of categorization, J. Improvements was about 10% with J = 3, 4, and about 5% with J = 2. Difference in improvement may be due to the difference in the denominator of the equations of GIDS and χ2. While GIDS has a normalizing denominator that has J in it, χ2 has an indefinite maximal end value that tends to grow as J increases. This may also increase the chance of overestimation that could reduce the performance.Comparison of the performance of GIDS-J2, -J3, and–J4 with the corresponding α, was taken as the type I error rate. Each data file in the null dataset was permuted by 1000 times to ensure the accuracy of the type I error, in percent, to one decimal place; α was set as 0.05. Estimation is reported in J = 2, 3, and 4 with each of the examined MAFs and heritability values. Maximum and minimum type I error rates in percent were , , and  for J = 2, 3, and 4, respectively, resulting in averages of 5.2, 5.2, and 5.3% for the corresponding J. Therefore it can be concluded that GIDS preserves the type I error rate over the conditions examined without any dependence on the number of outcome classes set for the analysis.The type I error rate was estimated using the null dataset generated following the same process as for simulation II, only without the causal pair. All the pairs among the 20 SNPs should not be found to have any noticeable association. The ratio of the permutation p-values smaller than or equal to a significance level, The Seventh Report of the Joint National Committee on Prevention, Detection, Evaluation, and Treatment of High Blood Pressure (JNC 7)  classifiJ was set as 9 in Eq (The Korean Association Resource (KARE) project  provided 9 in Eq . Then thWe propose a method to estimate genomic association and gene-gene interactions even when an outcome classification contains more than two classes. The lack of a mathematical limit on the number of outcome classes analyzable is mainly due to circumventing the explicit use of the concept of risk status. To apply the MDR method, we need to set the threshold for the discrimination of high and low risk. If only one threshold is required, as for binary outcomes, the odds ratio can be used to set a well-defined threshold value. But when considering an outcome with three or more classes, setting the multiple threshold values required would become increasingly ambiguous as the number of classes increases. Therefore, within the scheme of a confusion matrix, there may not be a straightforward way to measure the association between genotypic multifactor classes and multi-class phenotypes. The main advantage of using GIDS is the systematic extension of the number of analyzable outcome classes to arbitrary numbers by making the explicit use of the risk status unnecessary.J = 2.Many real phenotype data are observed as continuous variables in their intrinsic forms, which may be classified afterward and analysed as categorical variables. Classification may be strict or more inclusive. For example, for blood pressure, the detailed number of categories was changed from seven to six when the Seventh Report of the Joint National Committee on Prevention, Detection, Evaluation, and Treatment of High Blood Pressure (JNC7) was introduced, replacing JNC6 . ObesityAnother important contribution of GIDS is that multiple phenotypes can be analyzed as if they are a single composite trait. As can be seen in Although we have not used any cross-validation (CV), it can be easily implemented into our procedure. For example, a standard MDR procedure with CV can be proposed by using GIDS. If the best model is chosen based on the value of GIDS itself, then we need to consider overfitting. Instead of using CV, however, we proposed using a permutation approach based on randomization . By ‘ranIn summary, the proposed method could be applied in genomic association studies with a multi-class outcome either as a single phenotype or a multivariate composite phenotype.


Drosophila Genetic Reference Panel, a population of inbred, genome-sequenced lines derived from a natural population of Drosophila melanogaster, to investigate whether this population harbors genetic variation for a set of stress resistance and life history traits. Using a genomic approach, we found substantial genetic variation for metabolic rate, heat stress resistance, expression of a major heat shock protein, and egg-to-adult viability investigated at a benign and a higher stressful temperature. This suggests that these traits will be able to evolve. In addition, we outline an approach to conduct pathway associations based on genomic linear models, which has potential to identify adaptive genes and pathways, and therefore can be a valuable tool in conservation genomics.The ability of natural populations to withstand environmental stresses relies partly on their adaptive ability. In this study, we used a subset of the Understanding the genetic architecture of quantitative complex traits is a central topic in modern biology, with applications ranging within evolutionary genetics, animal and plant breeding, conservation biology, and human health. Linkage analyses and candidate gene studies have been used to gather information about the genetic basis of many complex phenotypes in a range of organisms, but usually with limited power to identify the causal loci. Linkage analyses rely on the joint inheritance of a small number of markers within families with known kinship. Candidate gene studies rely on a set of preselected genes; thus, many causal genes are likely to be missed because of incomplete genetic knowledge of the trait –3. With  Drosophila melanogaster, continue to play an important role in gaining increased knowledge about the genetic architecture of complex traits. Studies on this species have contributed to elucidating the genetic architecture of complex traits, including traits associated with stress resistance [ D. melanogaster Genetic Reference Panel (DGRP) [The principle of GWAS is to associate the phenotypic variation with genetic polymorphism (single-nucleotide polymorphisms (SNPs) and/or other polymorphic molecular variants) with the assumption that SNPs are in linkage disequilibrium (LD) with nearby causal variants or that the SNPs themselves are causal. For an increasing number of species, the genome has been sequenced, and combined with advances in bioinformatics, GWAS can be performed on model- as well as nonmodel organisms , 5. Howesistance , 7. The l (DGRP)  has ofteThe outcome of GWAS is typically a list of the most significant SNPs. This ignores much of the remaining variance due to the conservative statistical nature of such analyses. This is unfortunate because genetic variants with small effects are likely to be missed, and even variants with large effect may not be among the top hits , 9. AlthIncreasing evidence from studies relying on full genome approaches indicates that most traits have very complex genetic architectures and that they are influenced by often hundreds of interacting genes, each with a small effect on the phenotype and by genetic and environmental interactions –8, 12–15 D. melanogaster. We use a SNP-set approach in which we aggregate SNPs based on knowledge of biological processes (here gene ontologies (GOs)) and associate the phenotypic variation with genomic variation. We investigate whether these traits harbor genetic variation for five fitness-related phenotypes: egg-to-adult viability under benign and stressful temperatures, heat stress resistance, expression of a major heat shock protein (Hsp70), and metabolic rate. The capability of a genotype to produce an egg that successfully develops into an adult fly and the ability to withstand increasing stressful temperatures are key fitness traits of major importance for the abundance and distribution of insect species [In the present study we use a subset of the DGRP (21–27 lines depending on the trait assessed) to investigate five traits related to life history and environmental stress resistance in species . Metabol species –20 and r species . Knowled species . Drosophila Genetic Reference Panel (DGRP) [ Drosophila Stock Center (NIH P40OD018537) . The flies were maintained on a standard oatmeal-sugar-yeast-agar Drosophila medium at 25°C and a 12-hour light/dark cycle.A subset of thel (DGRP)  was obta2 was transformed. At benign and stressful temperatures 27 and 26 lines were assayed, respectively.The proportion of eggs which successfully develop to adulthood was assessed at two developmental temperatures: a benign temperature (25°C) and a lightly stressful temperature (28°C). For each line and temperature 200 eggs were collected. At benign temperature, forty eggs were placed in five vials per line, and at the high temperature twenty eggs were placed in ten vials per line. Emerging flies were counted daily until all flies had emerged. To approximate Gaussian distribution data arcsineApproximately 24 females from 27 lines were placed into individual glass tubes (24 × 15 mm) with lids. Tubes were randomized and placed on a rack that was submerged into a water bath heated to 37°C. Flies were constantly monitored after submersion, and heat stress resistance was measured as the time from placement in the water bath until the animal became comatose .2-production  and closed (45 minutes while CO2 accumulates). The system was controlled by two parallel 8-channel multiplexers . Opening allowed airflow of CO2-striped dry air  to flush the chambers at a fixed rate of 200 mL min−1 controlled by a mass flow controller  connected to a flow controller . Air leaving the metabolic chambers passed through a calcium chloride column  before entering the CO2 analyzer  to remove water. The temperature inside one metabolic chamber was registered by a data logger  and data were extracted by OneWireViewer . All flies were stored in a −80°C freezer after measurements and the dry weight  was obtained after drying for 24 hours at 60°C such that μL CO2 produced per hour per mg dry weight.The stop-flow respirometry system enabled analysis of the cumulative CO D. melanogaster [From each of 21 DGRP lines 10 five-day-old (±24 h) adult females were transferred to plastic tubes with screw cap and exposed to 35°C for 1 hour . Flies nogaster , using tnogaster . Prior t Wolbachia status were obtained from http://dgrp2.gnets.ncsu.edu/ [SNP data, major inversions, andcsu.edu/ , 28. They = Xb + Zg + ε, where y was a vector of phenotypic values, b was a vector of fixed effects t, In(2R)NS, In(3R)P, In(3R)K, and In(3R)Mo), and experimental block effects), g was a vector of random genomic effects, and ε was a vector of the residuals. X and Z were design matrices linking fixed and genomic effects to the observations. The genomic and residual effects were defined as g ~ N and ε ~ N. The additive genomic relationship matrix, G, was computed based on all genomic markers as G = WW′/m, where m was the total number of markers and W was a centered and scaled genotype matrix  = 0 and var(wi) = 1). Each column vector of W was pi was the minor allele frequency of the ith marker and ai was the ith column vector of the allele count matrix, A, containing the genotypes encoded as 0, 1, or 2, counting the number of minor alleles [Variance components and genomic effects were estimated using the REML algorithm implemented in the Regress package  for R 3 by fitti alleles .hSNP2 = σg2/(σg2 + σε2), and genomic (and raw phenotypic) correlations among the traits were computed as Spearman's rank correlation. The 95% confidence interval (CI) for hSNP2 was obtained using a bootstrap procedure; observations were sampled 10,000 times with replacement obtaining the same number of observations as the true data. In each round hSNP2 was estimated, and the 95% CI was then obtained as the 2.5% and 97.5% quantiles of the bootstrap samples.The genomic variance captured by common SNPs was computed as We used a pathway based approach to identify sets of SNPs with association with the traits. Firstly, SNPs were annotated to genes within a 5 KB region using variant annotation from FlyBase (v.FB5.46) . SecondlTsum), measuring the degree of association of one set of SNPs, was computed as the sum of the n marker effects  Randomly pick an element  was constructed. If the summary statistic for a SNP-set was below the threshold level (here p < 0.05) the corresponding element in the incidence matrix was set to 1, otherwise to 0. The observed overlap was then compared to an empirical distribution of the overlap. For a total of 10,000 times the elements within each column were permutated and the overlap among columns was recorded. The probability of the overlap was estimated under the null hypothesis of independent association among traits. We determined the empirical p value of a one-tailed test as the fraction of all random permutations that was larger than or equal to the observed overlap among traits at the 5% level.A consequence of the small number of lines investigated was limited statistical power to detect causal variants. Therefore, we investigated whether we observed shared patterns across traits in the rankings of the pathways. For each class of pathways  an incidence matrix with xj genes constituting the pathways  was computed as To dissect the genetic contribution of the associated pathways, the genetic variation within pathway was decomposed to gene level. Pathway-specific genetic effects of the 2 emission rate.In the present study we used a subset of the DGRP to investigate whether genetic variation existed for five fitness-related phenotypes, namely, the proportion of eggs that develop to adulthood at two environmental conditions, a benign and a mildly stressful temperature, resistance to acute heat stress, induction of a major heat shock protein (Hsp70), and metabolic rate estimated from COσg2) and a residual effect (σε2) from which the proportion of phenotypic variation captured by common SNPs was computed, that is, hSNP2. Metabolic rate, heat stress resistance, and Hsp70 expression all showed intermediate heritability estimates, whereas the two egg-to-adult viability traits both had high estimates , which were more than expected by chance. Three MF and one BP were associated with heat stress resistance, and three of each class were associated with Hsp70 expression (Table S4). Two or fewer GO within each class were associated with the egg-to-adult viability traits (Table S4). With exception of metabolic rate, the number of GOs below the threshold value was near the expected number of false-positives. Because of the apparent limited statistical power, we investigated whether we could identify general patterns of “association” across traits. Using a less conservative threshold level we computed overlaps of GOs with a p < 0.05 across traits , molecular functions MF, 239 GOs), and cellular components . With an arbitrary significance threshold of 39 GOs, a5 Table S, which w5 Table S, which w D. melanogaster population [ Drosophila [ Drosophila [Genetic variation is necessary for populations to adapt to variable and at times stressful environments. Using a model system, we found substantial phenotypic variation for the five traits related to fitness and environmental stress resistance . The DGRpulation ; thus oupulation . Howeverpulation , 38. Thepulation , therebyosophila , 41 but osophila , 43. Lasosophila  and diffosophila , 46. Oveosophila , 48, whiosophila . Why do osophila . It has osophila , which wWe found strong, significant negative genomic correlation between expression of a major heat shock protein and the two egg-to-adult viability traits and high positive genomic correlation between the two viability traits . Egg-to-Biological interpretation from associated GOs is difficult, especially with limited statistical power to detect pathways. However, in our study, we have shown an efficient method to compute individual markers' effects and subsequently a summary statistic for a set of markers. Despite limited statistical power, which we admittedly have in this study, some biological relevant pathways were identified. For example, GO:0000149 is a group of genes related to the SNARE complex, which was associated to heat stress resistance (Table S4). Several members of the heat shock protein family are implicated in the sustenance of synaptic function, and during synaptic transmission of vesicular content, for example, neurotransmitters, heat shock proteins bind to the SNARE complex . Another Frizzled, was in common among traits, between metabolic rate and Hsp70 expression, in this reduced list of genes  showed that most of the GOs had the same overall pattern, namely, relative few genes within each GO contributed to the overall genetic variation  and S6. of genes . Frizzleeostasis . One of  cascade . Also, ae stress , and Hspe stress .Despite the obvious statistical power limitations associated with the number of DGRP lines assessed in this study, we found more GOs than expected by chance, some of which seem to have important biological functions. More importantly, we found substantial genetic variation, thus evolutionary adaptive potential, for all five traits investigated supported by the relative narrow bootstrap CI for four of the traits.With the advances in sequencing technologies genomic approaches have been suggested as promising tools for conservation genetics. Neutral markers have predominately been applied in population and conservation genetics to describe loss of genetic variation, population structure, and so forth. However, using neutral markers to monitor the effect of environmental changes in a population is limited because the loss of variation will only decrease significantly if the population size is greatly reduced . TherefoIn the work presented here, we used a subset of the DGRP to investigate traits related to fitness and environmental stress resistance. As the environment changes, populations need to be able to adapt to these changes to survive. Using the DGRP we found substantial genetic variation for metabolic rate, heat stress resistance, expression of Hsp70, and egg-to-adult viability at two environmental conditions. In addition, we found evidence for genotype-by-environmental interaction for viability. Using a genomic pathways association approach, we attempted to locate pathways displaying association with the traits investigated. This approach can be extended to nonmodel organisms or provided as a genomic tool for identification of adaptive genes in model organisms and thus provide a potential use for conservation genomics.p<0.005 (Table S4) and GOs with a p<0.05 that are common across traits (Table S5). Lastly, the exact figures for Figure 2 are given in Table S6, that is, the associated GOs and the proportion of variance explained per gene adjusted for the number of SNPs within the gene.The supplementary material contain information of which DGRP lines assessed, additional experimental design informations (Table S1), and the phenotypic line means for the traits assayed (Table S2). The supplementary information also contains the likelihood ratio test for the fixed effects (Table S3), overview of GOs with a


In crossbreeding schemes, within-line selection of purebreds is performed mainly to improve the performance of crossbred descendants under field conditions. The genetic correlation between purebred and crossbred performance is an important parameter to be assessed because purebred performance can be a poor predictor of the performance of crossbred offspring. With the availability of high-density markers, the feasibility of using crossbred information to evaluate purebred candidates can be reassessed. This study implements and applies a single-step terminal-cross model (GEN) to real data to estimate the genetic parameters of several production and quality traits in pigs.longissimus dorsi muscle, drip loss and intramuscular fat content were recorded on all half-sibs. Animals were genotyped using the Illumina Porcine SNP60 BeadChip. The genetic correlation between purebred and crossbred performance was estimated separately for each trait. Purebred animals were evaluated using an animal model, whereas the additive genetic effect of a crossbred individual was decomposed into the additive effects of the sire and dam and a Mendelian sampling effect that was confounded with the residual effect. Genotypes of the Piétrain animals were integrated in the genetic evaluation by using a single-step procedure. As benchmarks, we used a model that was identical to GEN but only accounted for pedigree information (PED) and also two univariate single-step models (GEN_UNI) that took either purebred or crossbred performance into account.Piétrain sires were mated with Piétrain and Large White dams to produce purebred and crossbred male half-sib piglets; growth rate, feed conversion ratio, lean meat, pH of Genetic correlations between purebred and crossbred performance were high and positive for all traits (>0.69). Accuracies of estimated breeding values of genotyped sires and purebred offspring that were obtained with the GEN model outperformed both those obtained with the PED and the GEN_UNI models. The use of genomic information increased the predictive ability of the GEN model, but it did not substantially outperform the GEN_UNI models.We present a single-step terminal-cross model that integrates genomic information of purebred and crossbred performance by using available software. It improves the theoretical accuracy of genetic evaluations in breeding programs that are based on crossbreeding. The use of crossbreeding in breeding schemes reduces within-line inbreeding, and the resulting heterosis and breed complementarity have favorable effects on traits of commercial interest in crossbred (CB) animals. In crossbreeding schemes, individuals from the purebred (PB) parental lines are commonly selected in highly-sanitized environments in order to improve the performance of their CB descendants under field conditions, as a correlated response. In some cases, due to genetic and environmental differences between nucleus and field conditions, the performance of nucleus PB animals can be a poor predictor of the field performance of their CB descendants, which reduces the efficiency of the breeding program in terms of genetic progress at the commercial level . Thus, iThe theory on which selection for CB performance is based was largely developed and discussed many years ago , but untIn this study, we developed and tested on a real pig dataset a single-step terminal-cross model to estimate the genetic parameters of several production and quality traits. This model is based on the model of Wei and van der Werf , with anhttp://ethique.ipbs.fr/sdv/charteexpeanimale.pdf).The experiment was conducted according to the French guidelines for animal care and use  involved in the UtOpIGe project ANR-10-GENOM_BTV-015, which aims at investigating the feasibility of implementing genomic selection in pyramidal breeding schemes. A large number of traits that are difficult to measure under field conditions were recorded in several PB and CB populations that were raised in the same environment. In a first step, PB Piétrain and CB Piétrain × Large White animals were produced on selection and multiplication farms and tested at a single test station. In a second step, PB Piétrain sires were mated with sows of various CB and PB populations in order to test pigs that represent usual commercial production crosses at the same test station and to validate the results obtained from the first step.The present analyses involved only animals generated in the first step of the project, including 90 Piétrain boars (the offspring of 69 sires) and their descendants: 654 PB Piétrain and 716 CB Piétrain × Large White entire male piglets. Piétrain pigs are renowned for their very high yield of lean meat, whereas Large White pigs are reputed for their excellent maternal instinct, large litter size and high milk production. The PB and the CB descendants entered the test station facilities of Le Rheu (France) at approximately 5 weeks of age and were slaughtered at a fixed weight of 110 kg (at 5–6 months of age).longissimus dorsi muscle (pH), drip loss (DL), and intramuscular fat (IMF). Data were obtained in accordance with the national regulations on the welfare of animals used in research.The following traits were recorded: average daily gain from the beginning (35 kg) to the end (110 kg) of the test period (ADG), feed conversion ratio (FCR), % lean meat (LM), pH of the longissimus dorsi muscle was measured using a Xerolyt electrode  and a Sydel pH meter . DL was measured on a sample of loin of about 130 g (at the 13th lumbar vertebra). After weighing the samples, they were placed directly in a polystyrene tray, covered with polyethylene film and stored at 4 °C for 48 h, such that each slice formed a 40-degree angle with the horizontal plane. Then, the samples were wiped gently and weighed again. The DL was quantified as the difference between the two weights, expressed as a percentage of the initial weight. After DL measurement, the samples were frozen until IMF was measured by magnetic resonance imaging [At the slaughterhouse, carcasses were chilled in a cooling room at 4 °C for 24 h and right half-carcasses were cut . LM was  imaging .A separate pedigree file was constructed for each PB line. Piétrain and Large White pedigrees were constructed up to five generations back from the Piétrain boars for which data were available as PB or sires of CB offspring and from the Large White sows for which data were available as dams of CB offspring, respectively.The 90 Pietrain boars and their PB descendants were genotyped using the Illumina Porcine SNP60 BeadChip . SNPs with a call rate lower than 0.90 and a minor allele frequency lower than 0.05 were removed. For the remaining SNPs, the very few missing genotypes were imputed using a naïve method that sampled the genotypes with probability weights based on allele frequencies at each locus. Animals with a call rate lower than 0.90 and progeny that displayed Mendelian inconsistencies with their parents were discarded. Summary statistics of the phenotypes, pedigrees and genotypes are in Tables PB and CB phenotypes for a trait were considered as two different traits. They were analyzed jointly by adapting the terminal-cross model proposed by Wei and van der Werf (see Appendix 2 in ) by eithk = A,) and CB Piétrain × Large White individuals (for k = C), In matrix notation, the PB and CB records of a given trait in a pedigree-based terminal-cross model (PED) can be represented as follows:The Mendelian sampling effects cannot be estimated and are, therefore, included in the residual effect of the CB part of the model Eq. . Note thThe variance–covariance structure of additive genetic effects was assumed to follow:The form of the single-step terminal-cross model (GEN) is the same as that of Eq. . In the g values  is replamentclasspt{minima animals :\documenhttp://nce.ads.uga.edu/wiki/lib/exe/fetch.php?media=blupf90_all1.pdf). First, Van Raden’s equation [n × p , qj is the frequency of the minor allele of the jth SNP. Each column of matrix For PB Piétrain individuals (i.e. PB descendants and their sires), equation  was usedThen, n et al. . FinallySingle-step univariate models (GEN_UNI) were also run separately for PB and CB phenotypes by including the same effects as for the GEN model:A Bayesian framework was adopted for inference to express uncertainty about the unknowns through the use of probability density functions. Flat prior distributions were assumed for the parameters of the systematic effects and the (co)variance components. The Gibbs sampler algorithm was used to estimate the marginal posterior distributions of the systematic effects and the (co)variance components using the GIBBS1f90 software developed by Misztal et al. . Single σA(AC)* = 2σA(AC). The same rescaling also applies for the genetic variance on the maternal side.Because CB offspring performance includes only half of the breeding values for CB performance of their PB parents, Wei and van der Werf  assignedmentclass2pt{minimith individual for the kth  performance with the mth model  was calculated as [PEVi,k,m is the prediction error variance of its EBVi,k,m, Fi is the inbreeding coefficient of individual i, which was computed from the pedigree using the INBUPGf90 software [σk2 is the additive genetic variance of PB or CB performance. Theoretical accuracies of EBV were obtained for all sires and PB offspring and also for some PB descendants that were considered to be candidates for selection . Theoretical accuracies of EBV for these candidates were obtained using BLUP by either masking or including their own phenotype, in order to reproduce a situation under commercial conditions where some traits are measured on candidates and other traits are measured on relatives that are housed in test stations.The posterior mean (co)variance components obtained with the Gibbs sampler for the GEN model were used as the true values to obtain best linear unbiased prediction (BLUP) estimates for the breeding values in the different models using the BLUPf90 software . The thelated as :\documensoftware , and σk2The ability of the models to predict yet-to-be observed phenotypes was compared using sixfold cross-validation. First, BLUP solution estimates for the systematic effects obtained with the GEN model were used to obtain adjusted phenotype records, separately for each trait. Second, sires were randomly split into six approximately equal subsets. All records of the offspring of a sire were assigned to its respective subset. BLUP parameters were estimated based on five of the six data subsets, referred to as the training set, and the predictive ability was assessed in the remaining dataset, which was used as the testing set and considered as the yet-to-be observed phenotypes of the candidate sires. The training–testing cycle was repeated six times by rotating the six subsets used for training and testing, with each subset used only once for testing and five times for training. The predictive ability of each model was evaluated using the average Pearson’s correlation between observed and predicted phenotypes of the testing sets.In addition, the EBV of sires for PB performance estimated with the GEN_UNI model were used to predict CB performance under the GEN_UNI CB model. This predictive ability was also assessed with the sixfold cross-validation mentioned above.Genetic parameter estimates, ratios of variance components, and ratios of phenotypic variances obtained with the GEN model are in Table 95 %) of the estimates included values below 0.30. This indicates that most of the genetic variance observed for those traits is due to additive genes with no relevant dominance gene action and, possibly, no differences in gene frequency between the two lines, i.e. there is no strong genetic interaction between the Piétrain and Large White breeds. The magnitude and sign of the estimated genetic correlation between PB and CB performance are keys to decide the best strategy to evaluate PB animals for CB performance [Estimated genetic correlations between PB Piétrain and CB performance of Piétrain sires were all positive and high and none of the highest 95 % posterior density intervals  usiformance , 13, or formance , althougformance . Geneticformance . Geneticformance .The ratios of genetic variance for CB performance for the sire and dam lines were of similar magnitude across the traits, although they tended to be slightly higher for the sire line for FCR, pH and IMF Table . Heritabion data  but thishA2σA2 and tAC2σC2 + tBC2σC2 for PB and CB performance, respectively; Table The additive genetic variances  obtained with the single-step terminal-cross model and the mean differences of these accuracies from those obtained with the two univariate single-step models (one for CB and the other for PB performance) across traits are in Table Table Routine genetic evaluations in Piétrain pigs (as well as in other PB pig sire lines) are usually performed with pedigree-based BLUP using phenotypes of selection candidates and, for some traits, using phenotypes recorded on relatives obtained from test stations. Although selection is implemented within PB lines, the ultimate aim is to improve CB performance under field conditions. If the genetic correlations between PB and CB performance differ from 1, incorporating CB and genomic information into the genetic evaluation of the PB lines can contribute to increase genetic gains.To the best of our knowledge, this is the first implementation of a single-step terminal-cross model using PB sire genotypes to model CB performance and its application to real data for a wide range of traits. Christensen et al.  developeWe proposed and applied on real data a single-step terminal-cross model that accounts for genomic information on PB individuals and uses CB performance to estimate genetic parameters of several production and quality traits in pigs. Accounting for PB and CB information, along with genomic information, improves the theoretical accuracy of genetic evaluations in breeding programs that are based on crossbreeding. Including genomic information increased predictive abilities compared to using pedigree information only, but the single-step terminal-cross model did not outperform the predictive performance of univariate single-step models for PB and CB performance. The implementation of the proposed single-step terminal-cross model is straightforward with available software but its use under field conditions needs to be further addressed in terms of predictive ability, genetic progress achieved, and costs.


Both common and rare genetic variants have been shown to contribute to the etiology of complex diseases. Recent genome-wide association studies (GWAS) have successfully investigated how common variants contribute to the genetic factors associated with common human diseases. However, understanding the impact of rare variants, which are abundant in the human population (one in every 17 bases), remains challenging. A number of statistical tests have been developed to analyze collapsed rare variants identified by association tests. Here, we propose a haplotype-based approach. This work inspired by an existing statistical framework of the pedigree disequilibrium test (PDT), which uses genetic data to assess the effects of variants in general pedigrees. We aim to compare the performance between the haplotype-based approach and the rare variant-based approach for detecting rare causal variants in pedigrees.Extensive simulations in the sequencing setting were carried out to evaluate and compare the haplotype-based approach with the rare variant methods that drew on a more conventional collapsing strategy. As assessed through a variety of scenarios, the haplotype-based pedigree tests had enhanced statistical power compared with the rare variants based pedigree tests when the disease of interest was mainly caused by rare haplotypes , and vice versa when disease was caused by rare variants acting independently. For most of other situations when disease was caused both by haplotypes with multiple rare alleles and by rare variants with similar effects, these two approaches provided similar power in testing for association.The haplotype-based approach was designed to assess the role of rare and potentially causal haplotypes. The proposed rare variants-based pedigree tests were designed to assess the role of rare and potentially causal variants. This study clearly documented the situations under which either method performs better than the other. All tests have been implemented in a software, which was submitted to the Comprehensive R Archive Network (CRAN) for general use as a computer program named rvHPDT. Genetic studies such as genome-wide association studies (GWAS) have typically relied on the common disease/common variant (CDCV) paradigm, and the GWAS approach has had its share of success with some genetic disorders. Although earlier analyses of GWAS data revealed that this approach can detect common variants with modest effects, however, only a small portion of these significant associations with common variants were subsequently found to be functional and these associations account for a small portion of the total heritability of genetic variations. On the other hand, in recent years, the role of rare genetic variants has received more attention due to the advent of next generation sequencing (NGS) technology. Indeed, some rare variants were found to play a causal role in human diseases, including psychiatric disorders. This led to the shift from the CDCV approach to a common disease-rare variant (CDRV) approach.Given this increased recognition that common disorders may in fact reflect the aggregated effects of many rare variants, genetic analysts investigating this new hypothesis have had to deal with a host of new challenges. The more affordable whole genome sequencing (WGS) technology demands new analytical tools to analyze these data.FBAT software (v2.0.4), which provides both un-weighted (FBAT-v0) and weighted sum tests (FBAT-v1) to analyze rare variants . More recently, Slatter et al. [It is worth noting that all the family-based association tests listed above are variant-based approaches. Nevertheless, haplotypes—which are combinations of alleles  at adjacent locations on a chromosome—has the potential to provide further insight into the structure of underlying linkage disequilibrium (LD), thus potentially yielding higher power in association studies investigating common and rare variants , 16. Forr et al.  reportedr et al. . TogetheAt the theoretical level, Lin and colleagues  pointed hPDT, maxH, maxV, hPDT-t and vPDT-t. The detailed mathematical expressions will be provided under Methods. Our main goal is to investigate the performance of haplotype and variants based tests in general pedigrees, which will allow us to address the question of whether or not haplotype-based collapsing tests would outperform the variant-based collapsing tests for detecting rare variants in human pedigrees under difference cases of scenarios.Overall, we developed five tests termed N be the total number of pedigrees. Let M be the total number of variants in the gene of interest, and H haplotypes formed are denoted by h1, h2, …, hH. For the pedigree i, let nT be the number of informative nuclear families and nS be the number of informative discordant sibships. Similar to PDT, hPDT considers the difference in the number of transmitted and untransmitted haplotypes from parents to affected siblings and the difference in the number of haplotypes between affected and unaffected siblings. For k-th haplotype, the hPDT statistic for pedigree i is thus defined as:Let h th haplotype in trio j, and j.Where hH is the most frequent haplotype, and for the rest of H − 1 haplotypes, the summary random variable for the pedigree i is defined as:Without loss of generality, assume wk is the weight for the haplotype k. We define a similar weight function as in Madsen and colleagues [where lleagues :3λ is the number of individuals in N families, and qh is the haplotype frequency of the k-th haplotype. Under the null hypothesis of no association, E(Di) = 0 for any pedigree structure. If N families are unrelated, under the null hypothesis of no associationwhere andThe above derivation is adopted from the material provided by the original authors of the PDT test .N unrelated families, the hPDT statistic is defined as:For which asymptotically follows a standard normal distribution under the null hypothesis of no association based (see Equation ).vPDT test  is similarly defined as the hPDT test; however vPDT simply combines the M rare variants using allelic frequencies determined weights while the hPDT test use haplotype frequencies determined weights.In spirit, the k-th haplotype, the PDT statistic could be constructed asFor maxH test is defined as the maximal value of Zk) B times to obtain Tb, b = 1, …,B. These can be regarded as a sample of size B from the distribution of T , we propose to adopt the standard permutation procedure for obtaining p-values of Equation ) under tmaxH and maxV tests. For a simulation study with R replicates, the above procedure will be time-consuming. Therefore, we propose the following steps to reduce the computational burden by pooling permutation samples from all replicates to form a joint sample from the null distribution. More specifically, for the rth replicate with observed test statistic Tr, we permuted the labels B times as above to obtain Tr,b, b = 1, …,B. Then the collection {Tr,b; r = 1, …,R, b = 1, …,B} can be regarded as a random sample of size B × R from the common null distribution, i.e., the distribution under the null hypothesis H0. Consequently, for r = 1, …,R,provides a permutated p-value for the rth replicate. Since the permutation samples are pooled across all replicates to form a sample from the null, B can be set to be much smaller than the situation when only one sample is analyzed. For example, if R = 1000 and the desired number of permutations for estimating the p-values is 10000, then we set B = 10 for each replicate. In other words, we do not need to perform 10000 permutations for each replicate to obtain the p-value. Instead, we only need to perform 10 permutations per replicate to achieve the same precision level. This pooling strategy is similar to what was proposed in Becker and Knapp [provides the permuted p-value for the nd Knapp .To adapt to the case of scenarios that both risk haplotypes (increasing risks) and protective haplotypes (decreasing risks) exist, one may benefit from estimating the direction of each haplotype before the weights are defined using Equation . To accoZk) corresponds to a type I error of 0.3 . We set μ = 1.04 to preserve more nonzero weights/haplotypes in the next step tests. In the second step, the hPDT-t statistic is defined as Equation . Two hundred nuclear families were randomly selected, and each family included two parents, two affected siblings, and one unaffected sibling. The total numbers of individuals were 1000.For the purpose of simulation, the forward evolutionary simulation tool ForSim was usedForsim, we randomly picked 20 Kb regions as the causal region; only rare variants were to be tested in our simulation within this causal region. Here, rare variants were defined as “rare” when its MAF was between 0.001 and 0.05.To assign affected status we followed a disease model similar to that described by Li and colleagues . Within P = 30 % when simulating phenotype data under the null hypothesis.To investigate type I error rates, we set the penetrance at 30% in order to obtain a sufficient number of pedigrees with affected members. Disease status was assigned according to d%  of the rare variants in the region of interest to be causal. Then, among these rare variants, we further assumed that r%  of them increased disease risk, whereas the remaining (100 – r)% decreased disease risk. Following methods outlined by Madsen and Browning [l to the overall genotype relative risk (GRR) was defined as:To evaluate the power, we randomly selected PAR is the population attributable risk set as 0.006 [h1, h2}, the disease status was assigned according to:where as 0.006 . For an f0 was the baseline penetrance and was fixed at 10% in our simulations, Mc was the number of causal SNPs among M SNPs in the region, hk,l was the l th causal allele in haplotype hk, and al was the rare allele of SNP l.in which R2 values (R2 is a measurement of LD and defined as squared correlation coefficient between pairs of loci) between variant i and variant j, where i, j = 1, …, 10. There were four pair-wise LD In keeping with the method of Li and Leal , we alsohk1hk2 haplotype in the pedigree was assigned to be a random variable according to the following trait model, logit{P(affected|hk1hk2)} = βk1 + βk2, where the logit function is defined as k1 and k2  indicate 15 haplotypes defined as above. βk  indicate the effect sizes in trait model, and β s were specified as (1) β1 = β4 = 3, β5 = β6 = β7 = β8 = − 3; (2) β1 = β4 = 3, β5 = β6 = 1, β7 = β8 = − 3; (3) β1 = β2 = β3 = β4 = 2, and other βs were zero in each setting. Under this three β settings, the cumulative effect sizes (βs) on ten rare variants were ,  and , respectively.Finally, the affection status of an individual with forsim and by the direct simulation under the special haplotype setting. Six tests investigated here were as follows, hPDT: haplotype-based pedigree disequilibrium test; hPDT-t: haplotype-based pedigree disequilibrium test with 30% training data; maxV: maximal rare variant test; vPDT: variant-based pedigree disequilibrium test; vPDT-t: rare variant based pedigree disequilibrium test with 30% training data. We evaluated the effects of the proportion of causal variants (d%) and investigated the effects of the proportion of variants that increase risks (r%), and also investigated the effects of LD on power.Type I error rates and analytical power associated with six tests were assessed using the nuclear families simulated both by hPDT-t and vPDT-t tests were slightly conservative at type I error rates of 0.035 and 0.041, respectively. For these two tests, the weight of a haplotype/variant was set to be zero if it failed to get a significant value of test statistic on training data set as shown in Equation . The proportion of variants that increased risks, r%, was fixed at 100%, which means that all variants within the causal variants increase disease risk. Table hPDT and vPDT provided more power than the other four tests, and the hPDT-t and vPDT-t tests were least powerful, and the performances of maxH and maxV tests fell in the middle.We further evaluated the analytical power and the effects of the proportion of causal variants (r%) were also investigated. The proportion of causal variants, d%, was fixed at 50%, which means that half of the rare variants in the causal region were causal. Similar to results presented in Table r%=1 and 0.8, the hPDT and vPDT provided highest power than the other four tests, and the hPDT-t and vPDT-t were least powerful, and the maxH and maxV tests were in the middle. However, when r%=0.5 and 0.2, the maxH and maxV tests were most powerful when the joint effects were masked by noise from either causal variants with opposite direction  or non-causal variants (50%). When r%=0.05, except the hPDT-t and vPDT-t tests obtained power of ~30%, all other four tests have no power to detect association under this condition.The effects of the proportion of variants that increase risks , since the cumulative effect sizes βs on each single rare variant were added to zero, the haplotype-based tests outperformed the rare variants based tests. Specially, maxH obtained the highest power (0.589) due to the gain from the first and fourth haplotype with significant positive effect sizes. While the hPDT and hPDT-t tests were less powerful due to the fact that the positive and negative effects carried by different variants may cancel each other out. The training data provided good estimates on the direction of weights due to strong effect of the first and fourth haplotypes, therefore, hPDT-t (power = 0.394) was more powerful than the hPDT (power = 0.114) test.The effects of LD were investigated under the haplotype setting with LD and the results were shown in Table β setting β1 = β4 = 3, β5 = β6 = 1, β7 = β8 = − 3), the cumulative effect sizes βs on the first two variants were greater than 3, maxH remained to be most powerful (0.541) though the maxV test gained similar power (0.505). As same as under the first β setting, the hPDT-t outperformed hPDT since the risk, protective and natural haplotypes were well predicted by the training data set (using Equation (Under the second Equation ) when haβ setting (β1 = β2 = β3 = β4 = 2), the cumulative effect sizes (βs) on the first four variants were 4, which were greater than the haplotype effect sizes. Therefore, the rare variant based tests gained more power than the haplotype-based approaches . Under this parameter setting, as all effect sizes were in same direction on both haplotypes and rare variants, the methods based on training data had not gain any power than the methods without training data (hPDT-t < hPDT and vPDT-t < vPDT on power). The maxH test obtained was similar power under three β settings.Under the third Based upon the simulation results as shown in Table Forsim to simulate pedigrees that could be considered to be close to the real human population. In a real population, the haplotype frequencies were small on those haplotypes with two or more than two rare variants. For instance, the haplotype frequency was less than 0.0004 for a haplotype with two rare variants with equal to MAF of 0.02 when there is no LD between these two rare variants (i.e. Hardy-Weinberg Equilibrium). Under this circumstance, both haplotypes and rare variants had comparable contribution to disease since the causal variants were selected randomly, therefore, the two approaches had similar power in testing for association as shown in Tables When we used hPDT approach, we would like to note that the simulated genetic models were conducted under a few optimistic assumptions, and these assumptions may inflate the estimated power. In reality, hunting for rare and causal variants is known as a difficult task. Multiple variants from different genes can contribute to the genetic etiology of complex diseases, and looking at one gene at a time may cause missingness of causal variants. This is viewed as a limitation of our study. In future studies, we will broaden the scope of our research efforts by applying gene enrichment approaches to identify relevant molecular pathways involved in the occurrence of human diseases in multiple affected individuals in pedigrees. The extent of identity by descent (IBD) sharing will direct us where to look for the causal variants.Having demonstrated the power of the From a biological standpoint, several haplotype mutations are more likely to alter amino acid coding and therefore lead to a greater joint effect influencing traits of interest than a single amino acid change caused by a single mutation . The avaFinally, we acknowledge that, ideally, the feasibility of a newly developed test should be assessed using a real sequencing dataset. However, due to the rarity of sizable collections of family genomes in related individuals, we were unable to compare the above six tests in a real dataset. As an alternative, we applied those methods to a gene-based data set which contains a good portion of rare variants (10%) generated by the Illumina iSelect IBC Chip.hPDT, we evaluated the performance of our test using the Framingham Heart Study (FHS) data set (downloaded from dbGAP study: phs000007.v20.p8) and we also used the vPDT to conduct the analysis on the same data set. The FHS cohort includes 6911 individuals  in three generations; their mean and standard deviation (sd) age is 40.9 and 9.1, respectively. Forty-seven thousand and thirty-seven SNPs were genotyped in the Illumina iSelect IBC Chip. After quality control, there remained 4293 rare SNPs with MAFs≤0.05 in 780 genes in the FHS cohort, and we analyzed these SNPs using gene-based methods. The qualitative trait of hypertension was defined as systolic blood pressure (SBP) ≥ 140 or diastolic blood pressure (DBP) ≥ 90 or taking medicine for hypertension.To prove the feasibility of the hPDT-t and vPDT-t tests, the p-values were taken as 1 when all single haplotype/variant in training data set failed to pass the association test and all weights are set to be zero. Based upon the p-values shown in Table As noted in Table As shown in Table Here, we introduced a few novel haplotype-based PDT approaches for detecting rare variants using a weighted combination of rare haplotypes in pedigree data. As we reviewed earlier, in genetic studies, the haplotype structure is expected to carry important information. It has been found that the haplotype information gained from family members may result in more reliable estimates of the phase of haplotype. Moreover, as high-throughput sequencing is becoming widely used, the haplotyping methods can borrow some partial phase information in sequence reads to increase the accuracy of haplotypes estimated from genotype data .In addition, we sought to develop a method that could be used in family designs for following reasons: 1) family-based haplotype methods are more robust than case–control-based methods; and 2) if a rare haplotype is observed in a family, the haplotype may be enriched more frequently than in population-based designs. Therefore, the family-based haplotype design would be expected to be a more powerful of association between a collection of multiple rare variants and disease-relevant phenotypes.hPDT/hPDT-t and vPDT/vPDT-t approaches—drawn from the same PDT framework—were used in different sets if simulations. Further, maxH and maxV were also proposed to detect the signal of a single haplotype/variant. Such a comparison is the most intuitive way to assess whether the power increase are caused by the strategy used. Our simulations revealed that the haplotype-based approaches may achieve more power when the disease was more likely caused by causal haplotypes as shown in Table Both the maxH, hPDT, hPDT-t, maxV, vPDT and vPDT-t) have been implemented in the “rvHPDT” software coded by R language.To benefit potential users, all six tests (http://www.sph.umich.edu/csg/abecasis/merlin/tour/haplotyping.htmlMERLIN: http://cran.r-project.org/.The rvHPDT package has been submitted to the comprehensive R archive network (CRAN):


Principal component analysis (PCA) is a useful tool to identify important linear combination of correlated variables in multivariate analysis and has been applied to detect association between genetic variants and human complex diseases of interest. How to choose adequate number of principal components (PCs) to represent the original system in an optimal way is a key issue for PCA. Note that the traditional PCA, only using a few top PCs while discarding the other PCs, might significantly lose power in genetic association studies if all the PCs contain non-ignorable signals. In order to make full use of information from all PCs, Aschard and his colleagues have proposed a multi-step combined PCs method (named mCPC) recently, which performs well especially when several traits are highly correlated. However, the power superiority of mCPC has just been illustrated by simulation, while the theoretical power performance of mCPC has not been studied yet. In this work, we attempt to investigate theoretical properties of mCPC and further propose a novel and efficient strategy to combine PCs. Extensive simulation results confirm that the proposed method is more robust than existing procedures. A real data application to detect the association between gene TRAF1-C5 and rheumatoid arthritis further shows good performance of the proposed procedure. T2 test12489112Identification of genetic variants associated with human complex diseases can help investigators further understand genetic structure of diseases of interest. Compared with single-marker analysis, which tests every marker individually and is commonly employed in genome-wide association study, multiple-marker test has been well appreciated because of its potentially improved statistical power. Statistical methods for multiple-marker analysis can be summarized as synthesizing single-marker test statistics such as Hotelling’s et al.As it is well known that principal component analysis (PCA) is a useful tool to search for important characteristics among correlated variables. A key issue in developing an effective PCA model is choosing an adequate number of principal components (PCs) to represent the system in an optimal way. Taking advantage of the size of variances, HockingT follows a multivariate normal distribution with a m × 1 mean vector μ and known m × m covariance matrix V. We want to test the null hypothesis H0: μ = 0. Therefore, a Chi-squared statistic can be used for testing H0. However, when m is large, which is fairly common in genome-wide association studies, Chi-squared test might substantially lose power due to its large degrees of freedom. To reduce degrees of freedom, PCA is recommended. Based on orthogonal decomposition, we have V = QΛQτ, where qi is called as the eigenvector corresponding to the eigenvalue λi, Zi is related to the ith PC for H0, Without loss of generality, suppose a random vector m = 2, we consider a linear model with a normally distributed phenotype Y, which depends on two scaled genotypes G1 and G2 that are also normally distributed with mean 0 and variance 1. So the phenotype can be expressed as: Y = β0 + G1β1 + G2β2 + ε, where ε is the random error term which is distributed from the standard normal distribution. For this general model, the principal components of these two genotypes are Y can be reexpressed as: PC2 may be very important as PC2 is less than that of PC1. So we can not discard any PCs arbitrarily. In order to test H0: μ = 0, Aschard et al.Fk (·) is cumulative distribution function of a central Chi-squared random variable with k degrees of freedom. Moreover, they used simulation to compare the power of various PCA-based strategies when analyzing up to 100 correlated traits, and showed that their method with combining the signals across all PCs could have greater power. However, there has not been an in-depth study of the theoretical properties of mCPC in Aschard et al.’s paperet al.k. A commonly used method for selecting k is based on cumulative contribution rates, which are equal to ck for c ∈ . Aschard et al.k) with k being determined by cumulative contribution rate of 80%.For simplicity with In this work, we focus on the theoretical power of mCPC and find that the maximum power of mCPC is related to the maximum noncentral parameters under alternative hypothesis. We also find that the noncentral parameter corresponding to the top PC  is greater than 0 under most scenarios and those of other PCs do not possess this property when only a few means of all PCs are non-zero under alternative and the correlation coefficients among original variables are relatively large. Herein we propose a method tCPC. Based on numerical results, the tCPC is more powerful than the existing procedures under most of the considered scenarios.Ti is the statistic that is used to test for the association between the phenotype of interested and the ith genetic variants, V is the covariance matrix of the random vector T. Through the eigen-decomposition of the covariance matrix, we have V = QΛQτ, where qi is the eigenvector corresponding to the eigenvalue λi, H0, Zi follows a central Chi-squared distribution with 1 degree of freedom for i, For the multiple genetic variants association studies, the above random vector can be written as Fi (·). Note that for any given x ∈ , H0, both k) follows a central Chi-squared distribution with 4 degrees of freedom.For d degrees of freedom and non-centrality parameter ξ is x > 0, and According to Sankaranwhere with Cα1− be 1 − α quantile of a central Chi-squared distribution with 4 degrees of freedom. The power of mCPC (k) under the significance level α isLet Z1 which corresponds to the first PC is μ is the mean vector of T under the alternative hypothesis. Ω1 = 0 if and only if the mean vector μ belongs to the space that expanded by the other m − 1 eigenvectors m − 1 real numbers. However, for a m-dimensional space, Z1 is not equal to 0 almost everywhere. Besides this, and since the top PC possesses the largest variation among all PCs and k = 1 is a boundary point of the set consisting of Based on the above notations, the non-centrality parameter of the distribution of Under null hypothesis of no association at any locus, tCPC follows a central Chi-squared distribution with 4 degrees of freedom.T2 test (HT)12k0.8)In this subsection, we conduct simulation studies to compare powers between tCPC to some exiting approaches such as Hotelling’s m genetic variants (or SNPs) and a complex human disease. Let τ means the transpose of a vector or matrix. For example, we can construct T using the method in Chatterjee et al.m SNPs and a binary trait asConsider testing association between gi,j denotes the genotype of the jth SNP for the ith individual and n1 and n2 are the sample size of the case group and control group, respectively. Under the null hypothesis that these m SNPs are not associated with the disease of interest, T follows a multivariate normal distribution N  asymptotically with the mean vector μm×1 and covariance matrix VG is the pooled-sample covariance matrix of all SNPs.where T and Vm×m, we first generate a latent vector with length of 20 from a multivariate normal distribution with covariance structures of a compound symmetry with equal pairwise correlation ρ. Then, this latent vector is dichotomized to yield a haplotype with predesignated minor allele frequency (MAF). We repeat the above process 100,000 times to form a large population. Without loss of generality, we designate the first SNP as disease-causal SNP with MAF being p, and other SNPs as noncausal SNPs with MAFs all being q. Both sizes of case samples and control samples are set to be 1,000. Case or control status of one subject is generated from a logistic regression modelIn order to obtain β0 = −1.5, β1 ∈ {ln (1.4), ln (1.6)} denoting the log odds ratio for the disease-causal SNP, and Yi = 1 or 0 represents disease or healthy status of the ith individual, ρ = 0.50, β1 = 0 and p = q = 0.20, the empirical type I error rates of HT, oPC (k0.8), SSU, SKAT, mCPC (k0.8), and tCPC are 0.051, 0.052, 0.043, 0.049, 0.051, and 0.045, respectively. The results of power comparison shows that tCPC performs more robustly than the other methods. For example, when the correlation coefficients of these 20 SNPs are uniformly equal to 0.20 and β1 = ln 1.4, p = q = 0.20, the empirical powers of HT, oPC (k0.8), SSU, SKAT, mCPC (k0.8), and tCPC are 0.754, 0.736, 0.772, 0.766, 0.700, and 0.766, respectively. The empirical power of tCPC is a little lower than that of SSU in this scenario. However, when ρ = 0.50, β1 = ln 1.4, and p = q = 0.20, the empirical powers of HT, oPC (k0.8), SSU, SKAT, mCPC (k0.8), and tCPC are 0.724, 0.756, 0.810, 0.816, 0.702, and 0.856, respectively. It is obvious that tCPC performs the best among all the considered procedures in this setting.with ρi−j||)20×20. Other simulation settings are similar as above and are shown in ρ = 0.50, β1 = 0 and p = q = 0.20, the empirical type I error rates of HT, oPC (k0.8), SSU, SKAT, mCPC (k0.8), and tCPC are 0.036, 0.032, 0.04, 0.044, 0.038, and 0.042, respectively. For power comparison, tCPC still performs more robustly than the other methods. For example, when correlations of these 20 SNPs are decreasing with distance as ρ = 0.80, β1 = ln 1.4, p = 0.20, and q = 0.20, the empirical powers of HT, oPC (k0.8), SSU, SKAT, mCPC (k0.8), and tCPC are 0.77, 0.846, 0.772, 0.762, 0.788, and 0.836, respectively. The empirical power of tCPC is a little lower than that of oPC (k0.8) in this case. However, when ρ = 0.95, β1 = ln 1.4, p = 0.20, and q = 0.20, the powers of HT, oPC (k0.8), SSU, SKAT, mCPC (k0.8) and tCPC are 0.746, 0.898, 0.858, 0.858, 0.86 and 0.864, respectively. It indicates that tCPC gives the maximum power among the six methods in this scenario. Compared Next we consider a decreasing correlation structure. As a preliminary step, a latent vector with length of 20 is generated from a multivariate normal distribution with covariance matrix being (k0.8) and tCPC of detecting associations between gene TRAF1-C5 and rheumatoid arthritis are 5.21 × 10−5, 7.58 × 10−3, 5.95 × 10−4, 6.50 × 10−5, 7.56 × 10−5 and 3.75 × 10−5, respectively. If we use the p-value threshold of 5 × 10−5 as the moderate association at the genome-wide level as Burton et al.We apply tCPC and the other five existing tests to detect the association between gene TRAF1-C5 and rheumatoid arthritis using the data from the Genetic Analysis Workshop 16k0.8) and some other existing procedures such as Hotelling’s T2 test (HT), oPC (k0.8) SSU and SKAT by extensive simulations. All simulation results show that our proposed procedure is more robust than mCPC, HT, oPC (k0.8), SSU and SKAT. Results of real data analysis further demonstrates good performances of our proposed test. We suggest researchers to employ our robust strategy when they consider using principal component analysis method in the future.Principal component analysis is a common tool to grasp important features of correlated variables and has been applied in genetic association studies. In principal component analysis, cumulative contribution rate of 80% or 90% is commonly adopted to choose PCs. However, this adoption is not always suitable since PCs with low contribution rate might be much more strongly correlated with the outcome than those with large contribution rate. To overcome this drawback, a mCPC method was developed recentlyet al.It should be noted that our proposed procedure is built upon test by Chatterjee m, the powers of mCPC (k) mainly depend on k, μ and Vm×m. We set different mean vectors under the alternative hypothesis among different covariance matrices Vm×m. We also consider two types of Vm×m: one is that m-dimension variables are uniformly correlated, which means covariance matrix Vm×m is a symmetry positively definite matrix with diagonal elements all being 1 and non-diagonal elements all being ρ; the other is that all correlations among these m variables are decreasing considering the “physical” distance (SNP location), which means Vm×m = (ρi−j||)m×m. Without loss of generality, ρ is chosen to be 0.8 for strong linkage disequilibrium and 0.2 for weak linkage disequilibrium. Here, we consider m = 20. Note that, a test based on ordinary PCA can be gained, which is denoted by oPC (k) with k) are also affected by k when k) and oPC (k) comprehensively, we set α to be 0.05, and calculate powers of mCPC (k) and oPC (k) by numerical integration in R software under scenarios S1 to S16. All parameter settings about scenario S1 to S16 are displayed in Vm×m, ci, Ωi of all scenarios S1 to S16 for k) and oPC (k) are affected strongly by selection of k. From k) and oPC (k) are related to the maximum non-centrality parameters of all PCs as k is from 1 to 20. For example, in Scenario S15, the non-centrality parameter of the second PC is the maximum among those of all PCs, and mCPC (2) has the maximum power. In another example, under the scenario S2, the non-centrality parameter of the third PC is the maximum, mCPC (4) has the maximum power, and powers of mCPC (3) and mCPC (4) are close. We can also find out that ordinary way to select k to construct mCPC (k) is not always desirable. For example, in Scenario S12, k0.8 = 5, but oPC (5) has power as low as 0.115. It is verified that mCPC is more desirable than ordinary PCA based on k according to the cumulative contribution rate is not robust. One can just follow the common adoption and choose c = 80% or 90% for oPC (kc), but it will result in loss of power substantially under some situations. Furthermore, we draw a conclusion that mCPC (kc) performs more robust than oPC (kc) similar, since mCPC (kc) has reasonable powers over all the considered scenarios. For example, in Scenario S2, these 20 variables are uniformly correlated with ρ = 0.8 and k0.8) is 0.073, which is far less than the power of mCPC (kc), which is 0.57.For fixed  of mCPC  and mCPCum, mCPC  has the k) and oPC (k) shows that both of them are related to non-centrality parameters of the Chi-square distributions under the alternative hypothesis. For example, about scenario S16 in k become more prone to obtain. Aschard et al.k) with k being determined by cumulative contribution rate of 80%.A further investigation of the maximum powers of mCPC .