Partial least squares regression as an alternative to current regression methods used in ecology

This paper briefly presents the aims, requirements and results of partial least squares regression analysis (PLSR), and its potential utility in ecological studies. This statistical technique is particularly well suited to analyzing a large array of related predictor variables (i.e. not truly independent), with a sample size not large enough compared to the number of independent variables, and in cases in which an attempt is made to approach complex phenomena or syndromes that must be defined as a combination of several variables obtained independently. A simulation experiment is carried out to compare this technique with multiple regression (MR) and with a combination of principal component analysis and multiple regression (PCA+MR), varying the number of predictor variables and sample sizes. PLSR models explained a similar amount of variance to those results obtained by MR and PCA+MR. However, PLSR was more reliable than other techniques when identifying relevant variables and their magnitudes of influence, especially in cases of small sample size and low tolerance. Finally, we present one example of PLSR to illustrate its application and interpretation in ecology.     

Resistance to barley scald (Rhynchosporium secalis) in the Ethiopian donor lines 'Steudelli' and 'Jet', analyzed by partial least squares regression and interval mapping

The resistance of barley (Hordeum vulgare L.) to Rhynchosporium secalis (scald) has been investigated in two crosses between the susceptible cv. 'Ingrid' and two resistant Ethiopian landraces, 'Steudelli' and 'Jet'. Doubled haploids were inoculated in replicated tests using two isolates of R. secalis, '4004' and 'WRS1872'. Expression of resistance differed widely between replicated tests. AFLP, SSR and RFLP markers were used to develop chromosome maps. Results have been analysed using partial least squares regression (PLSR) and interval mapping. In PLSR the major covariance structures or 'latent variables' between X (markers) and Y (isolates, tests) are modelled as principal components and their optimal number determined by cross-validation. In 'Steudelli' two QTL were detected, one on each of chromosomes 3H and 7H, in 4 out of 5 tests, while in 'Jet' only one (different) allele at the 3H locus was found. The validated R(2) varied between 11.0% and 64.9% in the replicated tests with '4004'.With isolate 'WRS1872' the 7H locus and another 3H locus were detected. By interval mapping the QTL detected were less stable and generally gave lower R(2) values than PLSR. PLSR does not depend on maps, but interval mapping based on values predicted by PLSR had R(2) around 90%. It is suggested that PLSR may be a useful tool in QTL analysis.     

植物QTL分析的理论研究进展

数量性状的表型是由数量性状基因座 ( Quantitative trait locus,QTL)和环境效应共同作用的结果。传统的数量遗传学采用统计学的方法由一级统计量和二级统计量描述处理 QTL的复合作用 ,估计各种遗传参数 (例如遗传力、遗传相关、遗传进度、有效因子数等 ) ,用于指导遗传育种实践。然而 ,在传统的数量遗传学分析中 ,往往假设数量性状受微效多基因控制 ,这些基因具有相同的并且是较微小的效应 ,所估计的遗传参数反映的是数量性状多基因系统的整体特征 ,其理论方法不能用于追踪研究和描述单个数量性状基因的作用。近年来 ,由于分子生物学技…     

Interactions between markers can be caused by the dominance effect of quantitative trait loci

F₂ populations are commonly used in genetic studies of animals and plants. For simplicity, most quantitative trait locus or loci (QTL) mapping methods have been developed on the basis of populations having two distinct genotypes at each polymorphic marker or gene locus. In this study, we demonstrate that dominance can cause the interactions between markers and propose an inclusive linear model that includes marker variables and marker interactions so as to completely control both additive and dominance effects of QTL. The proposed linear model is the theoretical basis for inclusive composite-interval QTL mapping (ICIM) for F₂ populations, which consists of two steps: first, the best regression model is selected by stepwise regression, which approximately identifies markers and marker interactions explaining both additive and dominance variations; second, the interval mapping approach is applied to the phenotypic values adjusted by the regression model selected in the first step. Due to the limited mapping population size, the large number of variables, and multicollinearity between variables, coefficients in the inclusive linear model cannot be accurately determined in the first step. Interval mapping is necessary in the second step to fine tune the QTL to their true positions. The efficiency of including marker interactions in mapping additive and dominance QTL was demonstrated by extensive simulations using three QTL distribution models with two population sizes and an actual rice F₂ population.     

Multiple QTL mapping in related plant populations via a pedigree-analysis approach

QTL mapping experiments in plant breeding may involve multiple populations or pedigrees that are related through their ancestors. These known relationships have often been ignored for the sake of statistical analysis, despite their potential increase in power of mapping. We describe here a Bayesian method for QTL mapping in complex plant populations and reported the results from its application to a (previously analysed) potato data set. This Bayesian method was originally developed for human genetics data, and we have proved that it is useful for complex plant populations as well, based on a sensitivity analysis that was performed here. The method accommodates robustness to complex structures in pedigree data, full flexibility in the estimation of the number of QTL across multiple chromosomes, thereby accounting for uncertainties in the transmission of QTL and marker alleles due to incomplete marker information, and the simultaneous inclusion of non-genetic factors affecting the quantitative trait.     

Comparing linkage disequilibrium-based methods for fine mapping quantitative trait loci

Recently, a method for fine mapping quantitative trait loci (QTL) using linkage disequilibrium was proposed to map QTL by modeling covariance between individuals, due to identical-by-descent (IBD) QTL alleles, on the basis of the similarity of their marker haplotypes under an assumed population history. In the work presented here, the advantage of using marker haplotype information for fine mapping QTL was studied by comparing the IBD-based method with 10 markers to regression on a single marker, a pair of markers, or a two-locus haplotype under alternative population histories. When 10 markers were genotyped, the IBD-based method estimated the position of the QTL more accurately than did single-marker regression in all populations. When 20 markers were genotyped for regression, as single-marker methods do not require knowledge of haplotypes, the mapping accuracy of regression in all populations was similar to or greater than that of the IBD-based method using 10 markers. Thus for populations similar to those simulated here, the IBD-based method is comparable to single-marker regression analysis for fine mapping QTL.     

Approaches to interval mapping of QTL in a multigeneration pedigree: the example of porcine chromosome 4

Quantitative trait loci (QTLs) have been mapped in many studies of F2 populations derived from crosses between diverse lines. One approach to confirming these effects and improving the mapping resolution is genetic chromosome dissection through a backcrossing programme. Analysis by interval mapping of the data generated is likely to provide additional power and resolution compared with treating data marker by marker. However, interval mapping approaches for such a programme are not well developed, especially where the founder lines were outbred. We explore alternative approaches to analysis using, as an example, data from chromosome 4 in an intercross between wild boar and Large White pigs where QTLs have been previously identified. A least squares interval mapping procedure was used to study growth rate and carcass traits in a subsequent second backcross generation (BC2). This procedure requires the probability of inheriting a wild boar allele for each BC2 animal for locations throughout the chromosome. Two methods for obtaining these probabilities were compared: stochastic or deterministic. The two methods gave similar probabilities for inheriting wild boar alleles and, hence, gave very similar results from the QTL analysis. The deterministic approach has the advantage of being much faster to run but requires specialized software. A QTL for fatness and for growth were confirmed and, in addition, a QTL for piglet growth from weaning at 5 weeks up to 7 weeks of age and another for carcass length were detected.     

Interval mapping of quantitative trait loci with selective DNA pooling data

Selective DNA pooling is an efficient method to identify chromosomal regions that harbor quantitative trait loci (QTL) by comparing marker allele frequencies in pooled DNA from phenotypically extreme individuals. Currently used single marker analysis methods can detect linkage of markers to a QTL but do not provide separate estimates of QTL position and effect, nor do they utilize the joint information from multiple markers. In this study, two interval mapping methods for analysis of selective DNA pooling data were developed and evaluated. One was based on least squares regression (LS-pool) and the other on approximate maximum likelihood (ML-pool). Both methods simultaneously utilize information from multiple markers and multiple families and can be applied to different family structures (half-sib, F2 cross and backcross). The results from these two interval mapping methods were compared with results from single marker analysis by simulation. The results indicate that both LS-pool and ML-pool provided greater power to detect the QTL than single marker analysis. They also provide separate estimates of QTL location and effect. With large family sizes, both LS-pool and ML-pool provided similar power and estimates of QTL location and effect as selective genotyping. With small family sizes, however, the LS-pool method resulted in severely biased estimates of QTL location for distal QTL but this bias was reduced with the ML-pool.     

Identification of QTL for growth- and grain yield-related traits in rice across nine locations of Asia

Rice double-haploid (DH) lines of an indica and japonica cross were grown at nine different locations across four countries in Asia. Genotype-by-environment (G x E) interaction analysis for 11 growth- and grain yield-related traits in nine locations was estimated by AMMI analysis. Maximum G x E interaction was exhibited for fertility percentage number of spikelets and grain yield. Plant height was least affected by environment, and the AMMI model explained a total of 76.2% of the interaction effect. Mean environment was computed by averaging the nine environments and subsequently analyzed with other environments to map quantitative trait loci (QTL). QTL controlling the 11 traits were detected by interval analysis using mapmaker/qtl. A threshold LOD of >/=3.20 was used to identify significant QTL. A total of 126 QTL were identified for the 11 traits across nine locations. Thirty-four QTL common in more than one environment were identified on ten chromosomes. A maximum of 44 QTL were detected for panicle length, and the maximum number of common QTL were detected for days to heading detected. A single locus for plant height (RZ730-RG810) had QTL common in all ten environments, confirming AMMI results that QTL for plant height were affected the least by environment, indicating the stability of the trait. Two QTL were detected for grain yield and 19 for thousand-grain weight in all DH lines. The number of QTL per trait per location ranged from zero to four. Clustering of the QTL for different traits at the same marker intervals was observed for plant height, panicle number, panicle length and spikelet number suggesting that pleiotropism and or tight linkage of different traits could be the possible reason for the congruence of several QTL. The many QTL detected by the same marker interval across environments indicate that QTL for most traits are stable and not essentially affected by environmental factors.     

On the efficiency of composite interval mapping in an outbred population

Composite interval mapping (CIM) has been successfully applied to the detection of QTL in experimental animals and plants. However, practical analyses based on CIM have been reported mainly for populations derived from cross between inbred lines. There are few studies on QTL analyses with CIM in outbred populations. To evaluate the applicability of CIM to outbred populations is prerequisite for the fine mapping of QTL in industrial animals such as pig and chicken. Some markers are usually not fully informative in outbred populations. In application of CIM to outbred populations, the influence of inclusion of such uninformative markers used as covariates on the efficiency of CIM should be investigated. In this paper a least-squares method for CIM was formalized in an F(2) population derived by crossing two outbred lines. The efficiencies of CIM were evaluated for outbred populations in comparison with simple interval mapping (SIM) for several cases of marker informativeness using simulations. By incorporating markers linked to a tested position as well as those unlinked, CIM showed a higher efficiency to separate two linked QTL over SIM. The efficiency of dissection was enhanced as the marker informativeness was increased. The power of CIM to detect an isolated QTL was improved by excluding markers linked to a tested position from covariates and higher than SIM regardless of marker informativeness. In conclusion, CIM is a useful procedure for the analysis of QTL in outbred populations even under low marker informativeness.     

Robust indices of Hardy-Weinberg disequilibrium for QTL fine mapping

Hardy-Weinberg disequilibrium (HWD) measures have been proposed using dense markers to fine map a quantitative trait locus (QTL) to regions < approximately 1 cM. Earlier HWD measures may introduce bias in the fine mapping because they are dependent on marker allele frequencies across loci. Hence, HWD indices that do not depend on marker allele frequencies are desired for fine mapping. Based on our earlier work, here we present four new HWD indices that do not depend on marker allele frequencies. Two are for use when marker allele frequencies in a study population are known, and two are for use when marker allele frequencies in a study population are not known and are only known in the extreme samples. The new measures are a function of the genetic distance between the marker locus and a QTL. Through simulations, we investigated and compared the fine mapping performance of the new HWD measures with that of the earlier ones. Our results show that when marker allele frequencies vary across loci, the new measures presented here are more robust and powerful.     

Mixed-model QTL mapping for kernel hardness and dough strength in bread wheat

Plant breeding data comprise unbalanced phenotypic data for inbreds with complex pedigrees. As traditional methods to map quantitative trait loci (QTL) cannot exploit plant breeding data, an alternative approach is QTL mapping via a mixed-model procedure. Our objective was to validate mixed-model QTL mapping for self-pollinated crops by detecting QTL for kernel hardness and dough strength from data in a bread wheat (Triticum aestivum L.) breeding program. We studied 80 parental and 373 experimental inbreds genotyped for 65 simple sequence repeat (SSR) markers and three candidate loci. The methodology involved three steps: variance component estimation, single-marker analyses, and a final multiple-marker analysis with marker effects treated as fixed effects. Two QTLs for kernel hardness were detected on chromosomes 1A (close to candidate locus GluA3) and 5D (close to candidate locus Ha). Four QTLs were detected for dough strength on chromosomes 1A, 1B, 1D, and 5B. Candidate gene GluA1, which was associated with dough strength, was the only candidate locus found significant. Results were consistent with previously reported markers and QTLs associated with kernel hardness and dough strength. Unlike previous studies that have assumed QTL effects as random, the assumption of fixed marker effects identified the favorable marker alleles to select for. We conclude that the detection of previously mapped QTL validates the usefulness of mixed-model QTL mapping in the context of a plant-breeding program.     

Localization of QTLs for seed color using recombinant inbred lines of Brassica napus in different environments.

Yellow seed is one of the most important traits of Brassica napus L. Efficient selection of the yellow-seed trait is one of the most important objectives in oilseed rape breeding. Two recombinant inbred line (RIL) populations (RIL-1 and RIL-2) were analyzed for 2 years at 2 locations. Four hundred and twenty SSR, RAPD, and SRAP marker loci covering 1744 cM were mapped in 26 linkage groups of RIL-1, while 265 loci covering 1135 cM were mapped in 20 linkage groups of RIL-2. A total of 19 QTLs were detected in the 2 populations. A major QTL was detected adjacent to the same marker (EM11ME20/200) in both maps in both years. This major QTL could explain 53.71%, 39.34%, 42.42%, 30.18%, 24.86%, and 15.08% of phenotypic variation in 6 combinations (location x year x population). BLASTn analysis of the sequences of the markers flanking the major QTL revealed that the homologous region corresponding to this major QTL was anchored between genes At5g44440 and At5g49640 of Arabidopsis thaliana chromosome 5 (At C5). Based on comparative genomic analysis, the bifunctional gene TT10 is nearest to the homologue of EM11ME20/200 on At C5 and can be considered an important candidate gene for the major QTL identified here. Besides providing an effective strategy for marker-assisted selection of the yellow-seed trait in B. napus, our results also provide important clues for cloning of the candidate gene corresponding to this major QTL.     

Numerical comparison between powers of maximum likelihood and analysis of variance methods for QTL detection in progeny test designs: the case of monogenic inheritance

Simulations are used to compare four statistics for the detection of a quantitative trait locus (QTL) in daughter and grand-daughter designs as defined by Soller and Genizi (1978) and Weller et al. (1990): (1) the Fisher test of a linear model including a marker effect within sire or grand-sire effect; (2) the likelihood ratio test of a segregation analysis without the information given by the marker; (3) the likelihood ratio test of a segregation analysis considering the information from the marker; and (4) the lod score which is the likelihood ratio test of absence of linkage between the marker and the QTL. In all cases the two segregation analyses are more powerful for QTL detection than are either the linear method or the lod score. The differences in power are generally limited but may be significant (in a ratio of 1 to 3 or 4) when the QTL has a small effect (0.2 standard deviations) and is not closely linked to the marker (recombination rate of 20% or more).     

A two-phase procedure for QTL mapping with regression models

It is typical in QTL mapping experiments that the number of markers under investigation is large. This poses a challenge to commonly used regression models since the number of feature variables is usually much larger than the sample size, especially, when epistasis effects are to be considered. The greedy nature of the conventional stepwise procedures is well known and is even more conspicuous in such cases. In this article, we propose a two-phase procedure based on penalized likelihood techniques and extended Bayes information criterion (EBIC) for QTL mapping. The procedure consists of a screening phase and a selection phase. In the screening phase, the main and interaction features are alternatively screened by a penalized likelihood mechanism. In the selection phase, a low-dimensional approach using EBIC is applied to the features retained in the screening phase to identify QTL. The two-phase procedure has the asymptotic property that its positive detection rate (PDR) and false discovery rate (FDR) converge to 1 and 0, respectively, as sample size goes to infinity. The two-phase procedure is compared with both traditional and recently developed approaches by simulation studies. A real data analysis is presented to demonstrate the application of the two-phase procedure.     

Sensitivity of methods for estimating breeding values using genetic markers to the number of QTL and distribution of QTL variance

The objective of this simulation study was to compare the effect of the number of QTL and distribution of QTL variance on the accuracy of breeding values estimated with genomewide markers (MEBV). Three distinct methods were used to calculate MEBV: a Bayesian Method (BM), Least Angle Regression (LARS) and Partial Least Square Regression (PLSR). The accuracy of MEBV calculated with BM and LARS decreased when the number of simulated QTL increased. The accuracy decreased more when QTL had different variance values than when all QTL had an equal variance. The accuracy of MEBV calculated with PLSR was affected neither by the number of QTL nor by the distribution of QTL variance. Additional simulations and analyses showed that these conclusions were not affected by the number of individuals in the training population, by the number of markers and by the heritability of the trait. Results of this study show that the effect of the number of QTL and distribution of QTL variance on the accuracy of MEBV depends on the method that is used to calculate MEBV.     

Identification of Quantitative Trait Loci Influencing Wood Specific Gravity in an Outbred Pedigree of Loblolly Pine

We report the identification of quantitative trait loci (QTL) influencing wood specific gravity (WSG) in an outbred pedigree of loblolly pine (Pinus taeda L.). QTL mapping in an outcrossing species is complicated by the presence of multiple alleles (>2) at QTL and marker loci. Multiple alleles at QTL allow the examination of interaction among alleles at QTL (deviation from additive gene action). Restriction fragment length polymorphism (RFLP) marker genotypes and wood specific gravity phenotypes were determined for 177 progeny. Two RFLP linkage maps were constructed, representing maternal and paternal parent gamete segregations as inferred from diploid progeny RFLP genotypes. RFLP loci segregating for multiple alleles were vital for aligning the two maps. Each RFLP locus was assayed for cosegregation with WSG QTL using analysis of variance (ANOVA). Five regions of the genome contained one or more RFLP loci showing differences in mean WSG at or below the P = 0.05 level for progeny as grouped by RFLP genotype. One region contained a marker locus (S6a) whose QTL-associated effects were highly significant (P > 0.0002). Marker S6a segregated for multiple alleles, a prerequisite for determining the number of alleles segregating at the linked QTL and analyzing the interactions among QTL alleles. The QTL associated with marker S6a appeared to be segregating for multiple alleles which interacted with each other and with environments. No evidence for digenic epistasis was found among the five QTL.     

Detection and parameter estimation for quantitative trait loci using regression models and multiple markers

A strategy of multi-step minimal conditional regression analysis has been developed to determine the existence of statistical testing and parameter estimation for a quantitative trait locus (QTL) that are unaffected by linked QTLs. The estimation of marker-QTL recombination frequency needs to consider only three cases: 1) the chromosome has only one QTL, 2) one side of the target QTL has one or more QTLs, and 3) either side of the target QTL has one or more QTLs. Analytical formula was derived to estimate marker-QTL recombination frequency for each of the three cases. The formula involves two flanking markers for case 1), two flanking markers plus a conditional marker for case 2), and two flanking markers plus two conditional markers for case 3). Each QTL variance and effect, and the total QTL variance were also estimated using analytical formulae. Simulation data show that the formulae for estimating marker-QTL recombination frequency could be a useful statistical tool for fine QTL mapping. With 1 000 observations, a QTL could be mapped to a narrow chromosome region of 1.5 cM if no linked QTL is present, and to a 2.8 cM chromosome region if either side of the target QTL has at least one linked QTL.