On the use of permutation in and the performance of a class of nonparametric methods to detect differential gene expression

MOTIVATION: Recently a class of nonparametric statistical methods, including the empirical Bayes (EB) method, the significance analysis of microarray (SAM) method and the mixture model method (MMM), have been proposed to detect differential gene expression for replicated microarray experiments conducted under two conditions. All the methods depend on constructing a test statistic Z and a so-called null statistic z. The null statistic z is used to provide some reference distribution for Z such that statistical inference can be accomplished. A common way of constructing z is to apply Z to randomly permuted data. Here we point our that the distribution of z may not approximate the null distribution of Z well, leading to possibly too conservative inference. This observation may apply to other permutation-based nonparametric methods. We propose a new method of constructing a null statistic that aims to estimate the null distribution of a test statistic directly. RESULTS: Using simulated data and real data, we assess and compare the performance of the existing method and our new method when applied in EB, SAM and MMM. Some interesting findings on operating characteristics of EB, SAM and MMM are also reported. Finally, by combining the idea of SAM and MMM, we outline a simple nonparametric method based on the direct use of a test statistic and a null statistic.     

Theoretical and empirical power of regression and maximum-likelihood methods to map quantitative trait loci in general pedigrees

Both theoretical calculations and simulation studies have been used to compare and contrast the statistical power of methods for mapping quantitative trait loci (QTLs) in simple and complex pedigrees. A widely used approach in such studies is to derive or simulate the expected mean test statistic under the alternative hypothesis of a segregating QTL and to equate a larger mean test statistic with larger power. In the present study, we show that, even when the test statistic under the null hypothesis of no linkage follows a known asymptotic distribution (the standard being chi(2)), it cannot be assumed that the distribution under the alternative hypothesis is noncentral chi(2). Hence, mean test statistics cannot be used to indicate power differences, and a comparison between methods that are based on simulated average test statistics may lead to the wrong conclusion. We illustrate this important finding, through simulations and analytical derivations, for a recently proposed new regression method for the analysis of general pedigrees to map quantitative trait loci. We show that this regression method is not necessarily more powerful nor computationally more efficient than a maximum-likelihood variance-component approach. We advocate the use of empirical power to compare trait-mapping methods.     

Precision Mapping of Quantitative Trait Loci

Adequate separation of effects of possible multiple linked quantitative trait loci (QTLs) on mapping QTLs is the key to increasing the precision of QTL mapping. A new method of QTL mapping is proposed and analyzed in this paper by combining interval mapping with multiple regression. The basis of the proposed method is an interval test in which the test statistic on a marker interval is made to be unaffected by QTLs located outside a defined interval. This is achieved by fitting other genetic markers in the statistical model as a control when performing interval mapping. Compared with the current QTL mapping method (i.e., the interval mapping method which uses a pair or two pairs of markers for mapping QTLs), this method has several advantages. (1) By confining the test to one region at a time, it reduces a multiple dimensional search problem (for multiple QTLs) to a one dimensional search problem. (2) By conditioning linked markers in the test, the sensitivity of the test statistic to the position of individual QTLs is increased, and the precision of QTL mapping can be improved. (3) By selectively and simultaneously using other markers in the analysis, the efficiency of QTL mapping can be also improved. The behavior of the test statistic under the null hypothesis and appropriate critical value of the test statistic for an overall test in a genome are discussed and analyzed. A simulation study of QTL mapping is also presented which illustrates the utility, properties, advantages and disadvantages of the method.     

Evidence for a Language Quantitative Trait Locus on Chromosome 7q in Multiplex Autism Families

Autism is a syndrome characterized by deficits in language and social skills and by repetitive behaviors. We hypothesized that potential quantitative trait loci (QTLs) related to component autism endophenotypes might underlie putative or significant regions of autism linkage. We performed nonparametric multipoint linkage analyses, in 152 families from the Autism Genetic Resource Exchange, focusing on three traits derived from the Autism Diagnostic Interview: "age at first word," "age at first phrase," and a composite measure of "repetitive and stereotyped behavior." Families were genotyped for 335 markers, and multipoint sib pair linkage analyses were conducted. Using nonparametric multipoint linkage analysis, we found the strongest QTL evidence for age at first word on chromosome 7q (nonparametric test statistic [Z] 2.98; P=.001), and subsequent linkage analyses of additional markers and association analyses in the same region supported the initial result (Z=2.85, P=.002; chi(2)=18.84, df 8, P=.016). Moreover, the peak fine-mapping result for repetitive behavior (Z=2.48; P=.007) localized to a region overlapping this language QTL. The putative autism-susceptibility locus on chromosome 7 may be the result of separate QTLs for the language and repetitive or stereotyped behavior deficits that are associated with the disorder.     

Genotype-by-environment interaction in genetic mapping of multiple quantitative trait loci

The interval mapping method is widely used for the genetic mapping of quantitative trait loci (QTLs), though true resolution of quantitative variation into QTLs is hampered with this method. Separation of QTLs is troublesome, because single-QTL is models are fitted. Further, genotype-by-environment interaction, which is of great importance in many quantitative traits, can only be approached by separately analyzing the data collected in multiple environments. Here, we demonstrate for the first time a novel analytic approach (MQM mapping) that accommodates both the mapping of multiple QTLs and genotype-by-environment interaction. MQM mapping is compared to interval mapping in the mapping of QTLs for flowering time in Arabidopsis thaliana under various photoperiod and vernalization conditions.     

Power of in silico QTL mapping from phenotypic, pedigree, and marker data in a hybrid breeding program

Most quantitative trait locus (QTL) mapping studies in plants have used designed mapping populations. As an alternative to traditional QTL mapping, in silico mapping via a mixed-model approach simultaneously exploits phenotypic, genotypic, and pedigree data already available in breeding programs. The statistical power of this in silico mapping method, however, remains unknown. Our objective was to evaluate the power of in silico mapping via a mixed-model approach in hybrid crops. We used maize (Zea mays L.) as a model species to study, by computer simulation, the influence of number of QTLs (20 or 80), heritability (0.40 or 0.70), number of markers (200 or 400), and sample size (600 or 2,400 hybrids). We found that the average power to detect QTLs ranged from 0.11 to 0.59 for a significance level of =0.01, and from 0.01 to 0.47 for =0.0001. The false discovery rate ranged from 0.22 to 0.74 for =0.01, and from 0.05 to 0.46 for =0.0001. As with designed mapping experiments, a large sample size, high marker density, high heritability, and small number of QTLs led to the highest power for in silico mapping via a mixed-model approach. The power to detect QTLs with large effects was greater than the power to detect QTL with small effects. We conclude that gene discovery in hybrid crops can be initiated by in silico mapping. Finding an acceptable compromise, however, between the power to detect QTL and the proportion of false QTL would be necessary.     

QTL sequencing strategy to map genomic regions associated with resistance to ascochyta blight in chickpea

Whole‐genome sequencing‐based bulked segregant analysis (BSA) for mapping quantitative trait loci (QTL) provides an efficient alternative approach to conventional QTL analysis as it significantly reduces the scale and cost of analysis with comparable power to QTL detection using full mapping population. We tested the application of next‐generation sequencing (NGS)‐based BSA approach for mapping QTLs for ascochyta blight resistance in chickpea using two recombinant inbred line populations CPR‐01 and CPR‐02. Eleven QTLs in CPR‐01 and six QTLs in CPR‐02 populations were mapped on chromosomes Ca1, Ca2, Ca4, Ca6 and Ca7. The QTLs identified in CPR‐01 using conventional biparental mapping approach were used to compare the efficiency of NGS‐based BSA in detecting QTLs for ascochyta blight resistance. The QTLs on chromosomes Ca1, Ca4, Ca6 and Ca7 overlapped with the QTLs previously detected in CPR‐01 using conventional QTL mapping method. The QTLs on chromosome Ca4 were detected in both populations and overlapped with the previously reported QTLs indicating conserved region for ascochyta blight resistance across different chickpea genotypes. Six candidate genes in the QTL regions identified using NGS‐based BSA on chromosomes Ca2 and Ca4 were validated for their association with ascochyta blight resistance in the CPR‐02 population. This study demonstrated the efficiency of NGS‐based BSA as a rapid and cost‐effective method to identify QTLs associated with ascochyta blight in chickpea.     

The statistics of bulk segregant analysis using next generation sequencing

We describe a statistical framework for QTL mapping using bulk segregant analysis (BSA) based on high throughput, short-read sequencing. Our proposed approach is based on a smoothed version of the standard G statistic, and takes into account variation in allele frequency estimates due to sampling of segregants to form bulks as well as variation introduced during the sequencing of bulks. Using simulation, we explore the impact of key experimental variables such as bulk size and sequencing coverage on the ability to detect QTLs. Counterintuitively, we find that relatively large bulks maximize the power to detect QTLs even though this implies weaker selection and less extreme allele frequency differences. Our simulation studies suggest that with large bulks and sufficient sequencing depth, the methods we propose can be used to detect even weak effect QTLs and we demonstrate the utility of this framework by application to a BSA experiment in the budding yeast Saccharomyces cerevisiae.     

Zero-inflated generalized Poisson regression mixture model for mapping quantitative trait loci underlying count trait with many zeros

Phenotypes measured in counts are commonly observed in nature. Statistical methods for mapping quantitative trait loci (QTL) underlying count traits are documented in the literature. The majority of them assume that the count phenotype follows a Poisson distribution with appropriate techniques being applied to handle data dispersion. When a count trait has a genetic basis, “naturally occurring” zero status also reflects the underlying gene effects. Simply ignoring or miss-handling the zero data may lead to wrong QTL inference. In this article, we propose an interval mapping approach for mapping QTL underlying count phenotypes containing many zeros. The effects of QTLs on the zero-inflated count trait are modelled through the zero-inflated generalized Poisson regression mixture model, which can handle the zero inflation and Poisson dispersion in the same distribution. We implement the approach using the EM algorithm with the Newton-Raphson algorithm embedded in the M-step, and provide a genome-wide scan for testing and estimating the QTL effects. The performance of the proposed method is evaluated through extensive simulation studies. Extensions to composite and multiple interval mapping are discussed. The utility of the developed approach is illustrated through a mouse F₂ intercross data set. Significant QTLs are detected to control mouse cholesterol gallstone formation.     

基于性状—标记回归的QTL区间测验方法

本提两种基于性状－标记回归的QTL区间测验方法,分别称为TMRIT－I和TMRIT－II。前采用似然比统计量进行显性测验,与基于最小二乘的简化复合区间定位法（sCIM)等价,但计算机上明显简单快捷;后则采用一种“伪似然比”统计量进行显性测验,不仅进一步简化计算,而且明显提高统计功效,二皆可通过排列测验估计显阈值,给出了一个模拟例子。     

Bayesian multilocus association mapping on ordinal and censored traits and its application to the analysis of genetic variation among Oryza sativa L. germplasms

Association mapping can be a powerful tool for detecting quantitative trait loci (QTLs) without requiring line-crossing experiments. We previously proposed a Bayesian approach for simultaneously mapping multiple QTLs by a regression method that directly incorporates estimates of the population structure. In the present study, we extended our method to analyze ordinal and censored traits, since both types of traits are common in the evaluation of germplasm collections. Ordinal-probit and tobit models were employed to analyze ordinal and censored traits, respectively. In both models, we postulated the existence of a latent continuous variable associated with the observable data, and we used a Markov-chain Monte Carlo algorithm to sample the latent variable and determine the model parameters. We evaluated the efficiency of our approach by using simulated- and real-trait analyses of a rice germplasm collection. Simulation analyses based on real marker data showed that our models could reduce both false-positive and false-negative rates in detecting QTLs to reasonable levels. Simulation analyses based on highly polymorphic marker data, which were generated by coalescent simulations, showed that our models could be applied to genotype data based on highly polymorphic marker systems, like simple sequence repeats. For the real traits, we analyzed heading date as a censored trait and amylose content and the shape of milled rice grains as ordinal traits. We found significant markers that may be linked to previously reported QTLs. Our approach will be useful for whole-genome association mapping of ordinal and censored traits in rice germplasm collections.     

Bayesian association mapping of multiple quantitative trait loci and its application to the analysis of genetic variation among <Emphasis Type="Italic">Oryza sativa</Emphasis> L. germplasms

One way to use a crop germplasm collection directly to map QTLs without using line-crossing experiments is the whole genome association mapping. A major problem with association mapping is the presence of population structure, which can lead to both false positives and failure to detect genuine associations (i.e., false negatives). Particularly in highly selfing species such as Asian cultivated rice, high levels of population structure are expected and therefore the efficiency of association mapping remains almost unknown. Here, we propose an approach that combines a Bayesian method for mapping multiple QTLs with a regression method that directly incorporates estimates of population structure. That is, the effects due to both multiple QTLs and population structure were included in our statistical model. We evaluated the efficiency of our approach in simulated- and real-trait analyses of a rice germplasm collection. Simulation analyses based on real marker data showed that our model could suppress both false-positive and false-negative rates and the error of estimation of genetic effects over single QTL models, indicating that our model has statistically desirable attributes over single QTL models. As real traits, we analyzed the size and shape of milled rice grains and found significant markers that may be linked to QTLs reported previously. Association mapping should have good prospects in highly selfing species such as rice if proper methods are adopted. Our approach will be useful for the whole genome association mapping of various selfing crop species.     

Simultaneous confidence bands in spectral density estimation

We propose a method for the construction of simultaneous confidencebands for a smoothed version of the spectral density of a Gaussianprocess based on nonparametric kernel estimators obtained bysmoothing the periodogram. A studentized statistic is used todetermine the width of the band at each frequency and a frequency-domainbootstrap approach is employed to estimate the distributionof the supremum of this statistic over all frequencies. We proveby means of strong approximations that the bootstrap estimatesconsistently the distribution of the supremum deviation of interestand, consequently, that the proposed confidence bands achieveasymptotically the desired simultaneous coverage probability.The behaviour of our method in finite-sample situations is investigatedby simulations and a real-life data example demonstrates itsapplicability in time series analysis.     

QTL形态标记定位的一种数学方法

根据家蚕中位于Z染色体上的伴性遗传的双形态标记和假定与其有连锁关系的一个具有一对主基因差异的数量性状在测交世代中,所作的理论分布,本文建立了QTL形态标记定位的数学方法,即频数分布面积法,并给出了相应的检测一对主基因在测交世代中的同分离比例及其与形态标记是否有连锁关系的X2统计量．这种定位方法亦适应于非伴性遗传方式的QTL形态标记定位．与单标记定位的极大似然方法相比,我们的方法所作的双标记定位能显示QTL与形态标记发生重组的交叉干步作用,并且定位结果不受作用于数量性状的环境效应所影响．     

Nonparametric simulation-based statistics for detecting linkage in general pedigrees.

We present here four nonparametric statistics for linkage analysis that test whether pairs of affected relatives share marker alleles more often than expected. These statistics are based on simulating the null distribution of a given statistic conditional on the unaffecteds' marker genotypes. Each statistic uses a different measure of marker sharing: the SimAPM statistic uses the simulation-based affected-pedigree-member measure based on identity-by-state (IBS) sharing. The SimKIN (kinship) measure is 1.0 for identity-by-descent (IBD) sharing, 0.0 for no IBD status sharing, and the kinship coefficient when the IBD status is ambiguous. The simulation-based IBD (SimIBD) statistic uses a recursive algorithm to determine the probability of two affecteds sharing a specific allele IBD. The SimISO statistic is identical to SimIBD, except that it also measures marker similarity between unaffected pairs. We evaluated our statistics on data simulated under different two-locus disease models, comparing our results to those obtained with several other nonparametric statistics. Use of IBD information produces dramatic increases in power over the SimAPM method, which uses only IBS information. The power of our best statistic in most cases meets or exceeds the power of the other nonparametric statistics. Furthermore, our statistics perform comparisons between all affected relative pairs within general pedigrees and are not restricted to sib pairs or nuclear families.     

QTL analysis for transgressive resistance to root-knot nematode in interspecific cotton (Gossypium spp.) progeny derived from susceptible parents

The southern root-knot nematode (RKN, Meloidogyne incognita) is a major soil-inhabiting plant parasite that causes significant yield losses in cotton (Gossypium spp.). Progeny from crosses between cotton genotypes susceptible to RKN produced segregants in subsequent populations which were highly resistant to this parasite. A recombinant inbred line (RIL) population of 138 lines developed from a cross between Upland cotton TM-1 (G. hirsutum L.) and Pima 3-79 (G. barbadense L.), both susceptible to RKN, was used to identify quantitative trait loci (QTLs) determining responses to RKN in greenhouse infection assays with simple sequence repeat (SSR) markers. Compared to both parents, 53.6% and 52.1% of RILs showed less (P<0.05) root-galling index (GI) and had lower (P<0.05) nematode egg production (eggs per gram root, EGR). Highly resistant lines (transgressive segregants) were identified in this RIL population for GI and/or EGR in two greenhouse experiments. QTLs were identified using the single-marker analysis nonparametric mapping Kruskal-Wallis test. Four major QTLs located on chromosomes 3, 4, 11, and 17 were identified to account for 8.0 to 12.3% of the phenotypic variance (R(2)) in root-galling. Two major QTLs accounting for 9.7% and 10.6% of EGR variance were identified on chromosomes 14 and 23 (P<0.005), respectively. In addition, 19 putative QTLs (P<0.05) accounted for 4.5-7.7% of phenotypic variance (R(2)) in GI, and 15 QTLs accounted for 4.2-7.3% of phenotypic variance in EGR. In lines with alleles positive for resistance contributed by both parents in combinations of two to four QTLs, dramatic reductions of >50% in both GI and EGR were observed. The transgressive segregants with epistatic effects derived from susceptible parents indicate that high levels of nematode resistance in cotton may be attained by pyramiding positive alleles using a QTL mapping approach.     

Evaluation of linkage disequilibrium measures between multi-allelic markers as predictors of linkage disequilibrium between single nucleotide polymorphisms

Effectiveness of marker-assisted selection (MAS) and quantitative trait locus (QTL) mapping using population-wide linkage disequilibrium (LD) between markers and QTLs depends on the extent of LD and how it declines with distance between markers and QTLs in a population. Marker-QTL LD can be predicted from LD between markers. Our previous work evaluated LD measures between multi-allelic markers as predictors of usable LD of multi-allelic markers with QTLs. Since single nucleotide polymorphisms (SNPs) are the current marker of choice for high-density genotyping and LD-mapping of QTLs, the objective of this study was to use LD between multi-allelic markers to predict LD among biallelic SNPs or between SNPs and QTLs. Observable LD between multi-allelic markers was evaluated using nine measures. These included two pooled and standardized measures of LD between pairs of alleles at two markers based on Lewontin's LD measure, two pooled measures of squared correlations between alleles, one standardized measure using Hardy-Weinberg heterozygosities, and four measures based on the chi-square statistic for testing for association between alleles at two loci. The standardized chi-square measure that best predicted usable LD between multi-allelic markers and QTLs, based on our previous work, overestimated usable SNP-SNP or SNP-QTL LD. Instead, three other measures were found to be good predictors of usable SNP-SNP or SNP-QTL LD when LD is generated by drift. Therefore, the LD measure between multi-allelic markers that is best for predicting usable LD in a population depends on the type of markers (i.e. multi-allelic or biallelic) that will eventually be used for QTL mapping or MAS.     

A nonparametric bootstrap method for testing close linkage vs. pleiotropy of coincident quantitative trait loci.

A novel method using the nonparametric bootstrap is proposed for testing whether a quantitative trait locus (QTL) at one chromosomal position could explain effects on two separate traits. If the single-QTL hypothesis is accepted, pleiotropy could explain the effect on two traits. If it is rejected, then the effects on two traits are due to linked QTLs. The method can be used in conjunction with several QTL mapping methods as long as they provide a straightforward estimate of the number of QTLs detectable from the data set. A selection step was introduced in the bootstrap procedure to reduce the conservativeness of the test of close linkage vs. pleiotropy, so that the erroneous rejection of the null hypothesis of pleiotropy only happens at a frequency equal to the nominal type I error risk specified by the user. The approach was assessed using computer simulations and proved to be relatively unbiased and robust over the range of genetic situations tested. An example of its application on a real data set from a saline stress experiment performed on a recombinant population of wheat (Triticum aestivum L. ) doubled haploid lines is also provided.     

Controlling the Type I and Type II Errors in Mapping Quantitative Trait Loci

Although the interval mapping method is widely used for mapping quantitative trait loci (QTLs), it is not very well suited for mapping multiple QTLs. Here, we present the results of a computer simulation to study the application of exact and approximate models for multiple QTLs. In particular, we focus on an automatic two-stage procedure in which in the first stage ``important' markers are selected in multiple regression on markers. In the second stage a QTL is moved along the chromosomes by using the pre-selected markers as cofactors, except for the markers flanking the interval under study. A refined procedure for cases with large numbers of marker cofactors is described. Our approach will be called MQM mapping, where MQM is an acronym for ``multiple-QTL models' as well as for ``marker-QTL-marker.' Our simulation work demonstrates the great advantage of MQM mapping compared to interval mapping in reducing the chance of a type I error (i.e., a QTL is indicated at a location where actually no QTL is present) and in reducing the chance of a type II error (i.e., a QTL is not detected).     

Mapping QTLs with digenic epistasis under multiple environments and predicting heterosis based on QTL effects

Mixed linear model approach was proposed for mapping QTLs with the digenic epistasis and QTL by environment (QE) interaction as well as additive and dominant effects. Monte Carlo simulations indicated that the proposed method could provide unbiased estimations for both positions and genetic main effects of QTLs, as well as unbiased predictions for QE interaction effects. A method was suggested for predicting heterosis based on individual QTL effects. The immortalized F₂ (IF₂) population constructed by random mating among RI or DH lines is appropriate for mapping QTLs with epistasis and their QE interaction. Based on the models and methodology proposed, we developed a QTL mapping software, QTLMapper 2.0 on the basis of QTLmapper 1.0, which is suitable for analyzing populations of DH, RIL, F₂ and IF₂. Data of thousand grain weight of IF₂ population with 240 lines derived from elite hybrid rice Shanyou 63 were analyzed as a worked example.