A powerful and robust method for mapping quantitative trait loci in general pedigrees

The variance-components model is the method of choice for mapping quantitative trait loci in general human pedigrees. This model assumes normally distributed trait values and includes a major gene effect, random polygenic and environmental effects, and covariate effects. Violation of the normality assumption has detrimental effects on the type I error and power. One possible way of achieving normality is to transform trait values. The true transformation is unknown in practice, and different transformations may yield conflicting results. In addition, the commonly used transformations are ineffective in dealing with outlying trait values. We propose a novel extension of the variance-components model that allows the true transformation function to be completely unspecified. We present efficient likelihood-based procedures to estimate variance components and to test for genetic linkage. Simulation studies demonstrated that the new method is as powerful as the existing variance-components methods when the normality assumption holds; when the normality assumption fails, the new method still provides accurate control of type I error and is substantially more powerful than the existing methods. We performed a genomewide scan of monoamine oxidase B for the Collaborative Study on the Genetics of Alcoholism. In that study, the results that are based on the existing variance-components method changed dramatically when three outlying trait values were excluded from the analysis, whereas our method yielded essentially the same answers with or without those three outliers. The computer program that implements the new method is freely available.     

Quantitative similarity-based association tests using population samples

Although genetic association studies using unrelated individuals may be subject to bias caused by population stratification, alternative methods that are robust to population stratification, such as family-based association designs, may be less powerful. Furthermore, it is often more feasible and less expensive to collect unrelated individuals. Recently, several statistical methods have been proposed for case-control association tests in a structured population; these methods may be robust to population stratification. In the present study, we propose a quantitative similarity-based association test (QSAT) to identify association between a candidate marker and a quantitative trait of interest, through use of unrelated individuals. For the QSAT, we first determine whether two individuals are from the same subpopulation or from different subpopulations, using genotype data at a set of independent markers. We then perform an association test between the candidate marker and the quantitative trait, through incorporation of such information. Simulation results based on either coalescent models or empirical population genetics data show that the QSAT has a correct type I error rate in the presence of population stratification and that the power of the QSAT is higher than that of family-based association designs.     

Quantitative trait linkage analysis using Gaussian copulas

Mapping and identifying variants that influence quantitative traits is an important problem for genetic studies. Traditional QTL mapping relies on a variance-components (VC) approach with the key assumption that the trait values in a family follow a multivariate normal distribution. Violation of this assumption can lead to inflated type I error, reduced power, and biased parameter estimates. To accommodate nonnormally distributed data, we developed and implemented a modified VC method, which we call the "copula VC method," that directly models the nonnormal distribution using Gaussian copulas. The copula VC method allows the analysis of continuous, discrete, and censored trait data, and the standard VC method is a special case when the data are distributed as multivariate normal. Through the use of link functions, the copula VC method can easily incorporate covariates. We use computer simulations to show that the proposed method yields unbiased parameter estimates, correct type I error rates, and improved power for testing linkage with a variety of nonnormal traits as compared with the standard VC and the regression-based methods.     

Improving the power of sib pair quantitative trait loci detection by phenotype winsorization

OBJECTIVES: In sib pair studies, quantitative trait loci (QTL) identification may be adversely affected by non-normality in the phenotypic distribution, particularly when subjects falling in the tails of the distribution bias the trait mean or variance. We evaluated the robustness and power of reducing the influence of subjects with extreme phenotypic values by Winsorizing non-normal distributions in three versions of Haseman-Elston regression-based methods of QTL linkage analysis. METHODS: Data were simulated for normal and non-normal distributions. Phenotypic values that correspond to cutoff points at the omega and 1 - omega percentiles of the distribution were identified, and phenotypic values falling outside the boundaries of the omega and 1 - omega cutoff points were replaced by the omega and 1 - omega values, respectively. One million replications were performed for the three tests of linkage for Winsorized and non-Winsorized data. RESULTS: Winsorization reduced conservatism in the tails of the empirical type I error rate for the vast majority of the tests of linkage, increased the power of QTL detection in non-normal data and created a slight negative bias in symmetrical phenotypic distributions. CONCLUSIONS: Winsorizing can improve the power of QTL detection with certain non-normal distributions but can also introduce bias into the estimate of the QTL effect.     

Power studies in the estimation of genetic parameters and the localization of quantitative trait loci for backcross and doubled haploid populations

A statistical model for doubled haploids and backcrosses based on the interval-mapping methodology has been used to carry out power studies to investigate the effects of different experimental designs, heritabilities of the quantitative trait, and types of gene action, using two test statistics, the F of Fisher-Snedecor and the LOD score. The doubled haploid experimental design is more powerful than backcrosses while keeping actual type I errors similar to nominal ones. For the doubled haploid design, individual QTLs, showing heritabilities as low as 5% were detected in about 90% of the cases using only 250 individuals. The power to detect a given QTL is related to its contribution to the heritability of the trait. For a given nominal type I error, tests using F values are more powerful than with LOD scores. It seems that more conservative levels should be used for the LOD score in order to increase the power and obtain type I errors similar to nominal ones.     

Next generation modeling in GWAS: comparing different genetic architectures

The continuous advancement in genotyping technology has not been accompanied by the application of innovative statistical methods, such as multi-marker methods (MMM), to unravel genetic associations with complex traits. Although the performance of MMM has been widely explored in a prediction context, little is known on their behavior in the quantitative trait loci (QTL) detection under complex genetic architectures. We shed light on this still open question by applying Bayes A (BA) and Bayesian LASSO (BL) to simulated and real data. Both methods were compared to the single marker regression (SMR). Simulated data were generated in the context of six scenarios differing on effect size, minor allele frequency (MAF) and linkage disequilibrium (LD) between QTLs. These were based on real SNP genotypes in chromosome 21 from the Spanish Bladder Cancer Study. We show how the genetic architecture dramatically affects the behavior of the methods in terms of power, type I error and accuracy of estimates. Markers with high MAF are easier to detect by all methods, especially if they have a large effect on the phenotypic trait. A high LD between QTLs with either large or small effects differently affects the power of the methods: it impairs QTL detection with BA, irrespectively of the effect size, although boosts that of small effects with BL and SMR. We demonstrate the convenience of applying MMM rather than SMR because of their larger power and smaller type I error. Results from real data when applying MMM suggest novel associations not detected by SMR.     

The application of the entropy-based statistic for genomic association study of QTL

An entropy-based statistic TPE has been proposed for genomic association study for disease-susceptibility locus.The statistic TPE may be directly adopted and/or extended to quantitative-trait locus (QTL)mapping for quantitative traits.In this article,the statistic TPE was extended and applied to quantitative trait for association analysis of QTL by means of selective genotyping.The statistical properties (the type I error rate and the power) were examined under a range of parameters and population-sampling strategies (e.g.,various genetic models,various heritabilities,and various sample-selection threshold values) by simulation studies.The results indicated that the statistic Tee is robust and powerful for genomic association study of QTL.A simulation study based on the haplotype frequencies of 10 single nucleotide polymorphisms (SNPs) of angiotensin-I converting enzyme genes was conducted to evaluate the performance of the statistic TPE for genetic association study.     

Exact tests for the analysis of case-control studies of genetic markers

OBJECTIVES: The association of a candidate gene with disease can be evaluated by a case-control study in which the genotype distribution is compared for diseased cases and unaffected controls. Usually, the data are analyzed with Armitage's test using the asymptotic null distribution of the test statistic. Since this test does not generally guarantee a type I error rate less than or equal to the significance level alpha, tests based on exact null distributions have been investigated. METHODS: An algorithm to generate the exact null distribution for both Armitage's test statistic and a recently proposed modification of the Baumgartner-Weiss-Schindler statistic is presented. I have compared the tests in a simulation study. RESULTS: The asymptotic Armitage test is slightly anticonservative whereas the exact tests control the type I error rate. The exact Armitage test is very conservative, but the exact test based on the modification of the Baumgartner-Weiss-Schindler statistic has a type I error rate close to alpha. The exact Armitage test is the least powerful test; the difference in power between the other two tests is often small and the comparison does not show a clear winner. CONCLUSION: Simulation results indicate that an exact test based on the modification of the Baumgartner-Weiss-Schindler statistic is preferable for the analysis of case-control studies of genetic markers.     

Powerful regression-based quantitative-trait linkage analysis of general pedigrees

We present a new method of quantitative-trait linkage analysis that combines the simplicity and robustness of regression-based methods and the generality and greater power of variance-components models. The new method is based on a regression of estimated identity-by-descent (IBD) sharing between relative pairs on the squared sums and squared differences of trait values of the relative pairs. The method is applicable to pedigrees of arbitrary structure and to pedigrees selected on the basis of trait value, provided that population parameters of the trait distribution can be correctly specified. Ambiguous IBD sharing (due to incomplete marker information) can be accommodated in the method by appropriate specification of the variance-covariance matrix of IBD sharing between relative pairs. We have implemented this regression-based method and have performed simulation studies to assess, under a range of conditions, estimation accuracy, type I error rate, and power. For normally distributed traits and in large samples, the method is found to give the correct type I error rate and an unbiased estimate of the proportion of trait variance accounted for by the additive effects of the locus-although, in cases where asymptotic theory is doubtful, significance levels should be checked by simulations. In large sibships, the new method is slightly more powerful than variance-components models. The proposed method provides a practical and powerful tool for the linkage analysis of quantitative traits.     

New Insights into Handling Missing Values in Environmental Epidemiological Studies

Missing data are unavoidable in environmental epidemiologic surveys. The aim of this study was to compare methods for handling large amounts of missing values: omission of missing values, single and multiple imputations (through linear regression or partial least squares regression), and a fully Bayesian approach. These methods were applied to the PARIS birth cohort, where indoor domestic pollutant measurements were performed in a random sample of babies'' dwellings. A simulation study was conducted to assess performances of different approaches with a high proportion of missing values (from 50% to 95%). Different simulation scenarios were carried out, controlling the true value of the association (odds ratio of 1.0, 1.2, and 1.4), and varying the health outcome prevalence. When a large amount of data is missing, omitting these missing data reduced statistical power and inflated standard errors, which affected the significance of the association. Single imputation underestimated the variability, and considerably increased risk of type I error. All approaches were conservative, except the Bayesian joint model. In the case of a common health outcome, the fully Bayesian approach is the most efficient approach (low root mean square error, reasonable type I error, and high statistical power). Nevertheless for a less prevalent event, the type I error is increased and the statistical power is reduced. The estimated posterior distribution of the OR is useful to refine the conclusion. Among the methods handling missing values, no approach is absolutely the best but when usual approaches (e.g. single imputation) are not sufficient, joint modelling approach of missing process and health association is more efficient when large amounts of data are missing.     

Imprinting detection by extending a regression-based QTL analysis method

We present an extension of a regression-based quantitative-trait linkage analysis method to incorporate parent-of-origin effects. We separately regressed total, paternal, and maternal IBD sharing on traits’ squared sums and differences. We also developed a test for imprinting that indicates whether there is any difference between the paternal and maternal regression coefficients. Since this method treats the identity-by-descent information as the dependent variable that is conditioned on the trait, it can be readily applied to data from complex ascertainment processes. We performed a simulation study to examine the performance of the method. We found that when using empirical critical values, the method shows identical or higher power compared to existing methods for evaluation of parent-of-origin effect in linkage analysis of quantitative traits. Missing parental genotypes increase the type I error rate of the linkage test and decrease the power of the imprinting test. When the major gene has a low heritability, the power of the method decreases considerably, but the statistical tests still perform well. We also applied a permutation algorithm, which ensures the appropriate type I error rate for the test for imprinting. The method was applied to a data from a study of 6 body size related measures and 23 loci on chromosome 7 for 255 nuclear families. Multipoint identities-by-descent (IBD) were obtained using a modification of the SIMWALK 2 program. A parent-of-origin effect consistent with maternal imprinting was suggested at 99.67–111.26 Mb for body mass index, bioelectrical impedance analysis, waist circumference, and leptin concentration. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users. An erratum to this article can be found at     

Effect of Box-Cox transformation on power of Haseman-Elston and maximum-likelihood variance components tests to detect quantitative trait Loci

Non-normality of the phenotypic distribution can affect power to detect quantitative trait loci in sib pair studies. Previously, we observed that Winsorizing the sib pair phenotypes increased the power of quantitative trait locus (QTL) detection for both Haseman-Elston (HE) least-squares tests [Hum Hered 2002;53:59-67] and maximum likelihood-based variance components (MLVC) analysis [Behav Genet (in press)]. Winsorizing the phenotypes led to a slight increase in type 1 error in H-E tests and a slight decrease in type I error for MLVC analysis. Herein, we considered transforming the sib pair phenotypes using the Box-Cox family of transformations. Data were simulated for normal and non-normal (skewed and kurtic) distributions. Phenotypic values were replaced by Box-Cox transformed values. Twenty thousand replications were performed for three H-E tests of linkage and the likelihood ratio test (LRT), the Wald test and other robust versions based on the MLVC method. We calculated the relative nominal inflation rate as the ratio of observed empirical type 1 error divided by the set alpha level (5, 1 and 0.1% alpha levels). MLVC tests applied to non-normal data had inflated type I errors (rate ratio greater than 1.0), which were controlled best by Box-Cox transformation and to a lesser degree by Winsorizing. For example, for non-transformed, skewed phenotypes (derived from a chi2 distribution with 2 degrees of freedom), the rates of empirical type 1 error with respect to set alpha level=0.01 were 0.80, 4.35 and 7.33 for the original H-E test, LRT and Wald test, respectively. For the same alpha level=0.01, these rates were 1.12, 3.095 and 4.088 after Winsorizing and 0.723, 1.195 and 1.905 after Box-Cox transformation. Winsorizing reduced inflated error rates for the leptokurtic distribution (derived from a Laplace distribution with mean 0 and variance 8). Further, power (adjusted for empirical type 1 error) at the 0.01 alpha level ranged from 4.7 to 17.3% across all tests using the non-transformed, skewed phenotypes, from 7.5 to 20.1% after Winsorizing and from 12.6 to 33.2% after Box-Cox transformation. Likewise, power (adjusted for empirical type 1 error) using leptokurtic phenotypes at the 0.01 alpha level ranged from 4.4 to 12.5% across all tests with no transformation, from 7 to 19.2% after Winsorizing and from 4.5 to 13.8% after Box-Cox transformation. Thus the Box-Cox transformation apparently provided the best type 1 error control and maximal power among the procedures we considered for analyzing a non-normal, skewed distribution (chi2) while Winzorizing worked best for the non-normal, kurtic distribution (Laplace). We repeated the same simulations using a larger sample size (200 sib pairs) and found similar results.     

Robust genomic control for association studies

Population-based case-control studies are a useful method to test for a genetic association between a trait and a marker. However, the analysis of the resulting data can be affected by population stratification or cryptic relatedness, which may inflate the variance of the usual statistics, resulting in a higher-than-nominal rate of false-positive results. One approach to preserving the nominal type I error is to apply genomic control, which adjusts the variance of the Cochran-Armitage trend test by calculating the statistic on data from null loci. This enables one to estimate any additional variance in the null distribution of statistics. When the underlying genetic model (e.g., recessive, additive, or dominant) is known, genomic control can be applied to the corresponding optimal trend tests. In practice, however, the mode of inheritance is unknown. The genotype-based chi (2) test for a general association between the trait and the marker does not depend on the underlying genetic model. Since this general association test has 2 degrees of freedom (df), the existing formulas for estimating the variance factor by use of genomic control are not directly applicable. By expressing the general association test in terms of two Cochran-Armitage trend tests, one can apply genomic control to each of the two trend tests separately, thereby adjusting the chi (2) statistic. The properties of this robust genomic control test with 2 df are examined by simulation. This genomic control-adjusted 2-df test has control of type I error and achieves reasonable power, relative to the optimal tests for each model.     

基于选择基因型对数量性状进行关联分析

数量性状的遗传分析可以通过＂选择基因型＂的方式完成。本文提出了一个利用极端样本来对数量性状位点（QTL）进行关联分析的统计量T。统计量T比较上极端群体样本中具有纯合子标记的性状值差异。通过计算机模拟考察了无关联情形时T的分布和Ⅰ型错误率,结果表明,在各种样本选择策略下,T的分布近似于χ^2-分布,Ⅰ型错误率接近设定的显著性水平。同时,考察了各种遗传模型下不同遗传率,不同样本大小,及不同样本选择阈值对T的统计功效的影响,结果表明,T的功效随着标记和QTL间连锁不平衡程度的增强及遗传率和样本大小的增大而增大,当样本选择阈值更严格时,功效也越大。     

Trait response in communities to environmental change: effect of interspecific competition and trait covariance structure

The response of ecological communities to environmental disturbances depends not just on the number of species they contain but also on the functional diversity of the constituent species; greater variation in the tolerance of species to different environmental disturbances is generally thought to confer greater resistance to the community. Here, I investigate how the functional diversity of communities changes with environmental disturbances. Specifically, I assume that there is variation in traits among species that confer tolerance or sensitivity to environmental disturbances. When a disturbance occurs, variation in species tolerances causes changes in the relative abundances of species, which in turn changes the average tolerance of the community. For example, if tolerance to an environmental disturbance is conferred by large body size, then the environmental disturbance should be expected to increase the average body size of individuals in the community. Despite this expectation, ecological interactions among species can affect the average community response. For example, if larger species are also strong competitors with each other, then this might reduce the increase in average body size in the community, because interspecific competition limits the grow in population density of large bodied species. Similarly, when disturbances affect multiple traits, the covariance in the distribution of trait values among species may restrict the response of any one trait; if two traits provide tolerance to the same disturbance but negatively covary among species, then the response of one trait will limit the response of the other trait at the community level. Using a Lotka–Volterra model for competitive communities, I derive general formulae that generate explicit predictions about the changes in average trait values in a community subject to environmental disturbances. These formulae demonstrate that competition can impede the change in average community trait values. However, the impediment is not considerable in comparison to the predominant factors of trait variances and species selection effects when species with the most similar trait values also experience the greatest interspecific competition. Similarly, negative covariances among different traits that confer resistance to the same environmental disturbance will impede their responses. I illustrate these results using phytoplankton data from a whole-lake experiment in which manipulation to the zooplankton community created a disturbance to the phytoplankton that changed the selective consumption of large vs. small phytoplankton.     

Multiple trait multiple interval mapping of quantitative trait loci from inbred line crosses

ABSTRACT: BACKGROUND: Although many experiments have measurements on multiple traits, most studies performed the analysis of mapping of quantitative trait loci (QTL) for each trait separately using single trait analysis. Single trait analysis does not take advantage of possible genetic and environmental correlations between traits. In this paper, we propose a novel statistical method for multiple trait multiple interval mapping (MTMIM) of QTL for inbred line crosses. We also develop a novel score-based method for estimating genome-wide significance level of putative QTL effects suitable for the MTMIM model. The MTMIM method is implemented in the freely available and widely used Windows QTL Cartographer software. RESULTS: Throughout the paper, we provide compelling empirical evidences that: (1) the score-based threshold maintains proper type I error rate and tends to keep false discovery rate within an acceptable level; (2) the MTMIM method can deliver better parameter estimates and power than single trait multiple interval mapping method; (3) an analysis of Drosophila dataset illustrates how the MTMIM method can better extract information from datasets with measurements in multiple traits. CONCLUSIONS: The MTMIM method represents a convenient statistical framework to test hypotheses of pleiotropic QTL versus closely linked nonpleiotropic QTL, QTL by environment interaction, and to estimate the total genotypic variance-covariance matrix between traits and to decompose it in terms of QTL-specific variance-covariance matrices, therefore, providing more details on the genetic architecture of complex traits.     

Importance sampling method of correction for multiple testing in affected sib-pair linkage analysis

Using the Genetic Analysis Workshop 13 simulated data set, we compared the technique of importance sampling to several other methods designed to adjust p-values for multiple testing: the Bonferroni correction, the method proposed by Feingold et al., and na?ve Monte Carlo simulation. We performed affected sib-pair linkage analysis for each of the 100 replicates for each of five binary traits and adjusted the derived p-values using each of the correction methods. The type I error rates for each correction method and the ability of each of the methods to detect loci known to influence trait values were compared. All of the methods considered were conservative with respect to type I error, especially the Bonferroni method. The ability of these methods to detect trait loci was also low. However, this may be partially due to a limitation inherent in our binary trait definitions.     

Which strategy is better for linkage analysis: single-nucleotide polymorphisms or microsatellites? Evaluation by identity-by-state – identity-by-descent transformation affected sib-pair method on GAW14 data

The central issue for Genetic Analysis Workshop 14 (GAW14) is the question, which is the better strategy for linkage analysis, the use of single-nucleotide polymorphisms (SNPs) or microsatellite markers? To answer this question we analyzed the simulated data using Duffy's SIB-PAIR program, which can incorporate parental genotypes, and our identity-by-state – identity-by-descent (IBS-IBD) transformation method of affected sib-pair linkage analysis which uses the matrix transformation between IBS and IBD. The advantages of our method are as follows: the assumption of Hardy-Weinberg equilibrium is not necessary; the parental genotype information maybe all unknown; both IBS and its related IBD transformation can be used in the linkage analysis; the determinant of the IBS-IBD transformation matrix provides a quantitative measure of the quality of the marker in linkage analysis. With the originally distributed simulated data, we found that 1) for microsatellite markers there are virtually no differences in types I and II error rates when parental genotypes were or were not used; 2) on average, a microsatellite marker has more power than a SNP marker does in linkage detection; 3) if parental genotype information is used, SNP markers show lower type I error rates than microsatellite markers; and 4) if parental genotypes are not available, SNP markers show considerable variation in type I error rates for different methods.