A simple linear regression method for quantitative trait loci linkage analysis with censored observations

Standard quantitative trait loci (QTL) mapping techniques commonly assume that the trait is both fully observed and normally distributed. When considering survival or age-at-onset traits these assumptions are often incorrect. Methods have been developed to map QTL for survival traits; however, they are both computationally intensive and not available in standard genome analysis software packages. We propose a grouped linear regression method for the analysis of continuous survival data. Using simulation we compare this method to both the Cox and Weibull proportional hazards models and a standard linear regression method that ignores censoring. The grouped linear regression method is of equivalent power to both the Cox and Weibull proportional hazards methods and is significantly better than the standard linear regression method when censored observations are present. The method is also robust to the proportion of censored individuals and the underlying distribution of the trait. On the basis of linear regression methodology, the grouped linear regression model is computationally simple and fast and can be implemented readily in freely available statistical software.     

Multiple regression for molecular-marker,quantitative trait data from large F2 populations

Molecular marker-quantitative trait associations are important for breeders to recognize and understand to allow application in selection. This work was done to provide simple, intuitive explanations of trait-marker regression for large samples from an F₂ and to examine the properties of the regression estimators. Beginning with a(- 1,0,1) coding of marker classes and expected frequencies in the F₂, expected values, variances, and covariances of marker variables were calculated. Simple linear regression and regression of trait values on two markers were computed. The sum of coefficient estimates for the flanking-marker regression is asymptotically unbiased for an included additive effect with complete interference, and is only slightly biased with no interference and moderately close (15 cM) marker spacing. The variance of the sum of regression coefficients is much more stable for small recombination distances than variances of individual coefficients. Multiple regression of trait variables on coded marker variables can be interpreted as the product of the inverse of the marker correlation matrix R and the vector a of simple linear regression estimators for each marker. For no interference, elements of the correlation matrix R can be written as products of correlations between adjacent markers. The inverse of R is displayed and used to illustrate the solution vector. Only markers immediately flanking trait loci are expected to have non-zero values and, with at least two marker loci between each trait locus, the solution vector is expected to be the sum of solutions for each trait locus. Results of this work should allow breeders to test for intervals in which trait loci are located and to better interpret results of the trait-marker regression.     

Marker-assisted selection and marker-QTL associations in hybrid populations

A detailed analysis is presented of the relationship between genetic markers and quantitative trait loci (QTLs) in the process of marker-assisted selection (MAS). We simulated MAS employing a multiple linear regression to chose from among all of the markers in the genome those to be utilized by selection and to estimate their associated effects on the trait. The simulations demonstrate that, even when such selection is quite effective, the markers utilized by selection are not necessarily the most tightly linked to the QTLs controling the trait. Moreover, the additive effects associated with the markers estimated by the regression may not accurately reflect the contributions to the trait by the most tightly linked QTLs.     

Regression-based sib pair linkage analysis for binary traits

The Haseman-Elston (HE) regression method offers a mathematically and computationally simpler alternative to variance-components (VC) models for the linkage analysis of quantitative traits. However, current versions of HE regression and VC models are not optimised for binary traits. Here, we present a modified HE regression and a liability-threshold VC model for binary-traits. The new HE method is based on the regression of a linear combination of the trait squares and the trait cross-product on the proportion of alleles identical by descent (IBD) at the putative locus, for sibling pairs. We have implemented both the new HE regression-based method and have performed analytic and simulation studies to assess its type 1 error rate and power under a range of conditions. These studies showed that the new HE method is well-behaved under the null hypothesis in large samples, is more powerful than both the original and the revisited HE methods, and is approximately equivalent in power to the liability-threshold VC model.     

Uncovering a Nuisance Influence of a Phenological Trait of Plants Using a Nonlinear Structural Equation: Application to Days to Heading and Culm Length in Asian Cultivated Rice (Oryza Sativa L.)

Phenological traits of plants, such as flowering time, are linked to growth phase transition. Thus, phenological traits often influence other traits through the modification of the duration of growth period. This influence is a nuisance in plant breeding because it hampers genetic evaluation of the influenced traits. Genetic effects on the influenced traits have two components, one that directly affects the traits and one that indirectly affects the traits via the phenological trait. These cannot be distinguished by phenotypic evaluation and ordinary linear regression models. Consequently, if a phenological trait is modified by introgression or editing of the responsible genes, the phenotypes of the influenced traits can change unexpectedly. To uncover the influence of the phenological trait and evaluate the direct genetic effects on the influenced traits, we developed a nonlinear structural equation (NSE) incorporating a nonlinear influence of the phenological trait. We applied the NSE to real data for cultivated rice (Oryza sativa L.): days to heading (DH) as a phenological trait and culm length (CL) as the influenced trait. This showed that CL of the cultivars that showed extremely early heading was shortened by the strong influence of DH. In a simulation study, it was shown that the NSE was able to infer the nonlinear influence and direct genetic effects with reasonable accuracy. However, the NSE failed to infer the linear influence in this study. When no influence was simulated, an ordinary bi-trait linear model (OLM) tended to infer the genetic effects more accurately. In such cases, however, by comparing the NSE and OLM using an information criterion, we could assess whether the nonlinear assumption of the NSE was appropriate for the data analyzed. This study demonstrates the usefulness of the NSE in revealing the phenotypic influence of phenological traits.     

鼎湖山南亚热带常绿阔叶林叶功能性状沿群落垂直层次的种内变异

探究功能性状沿着环境梯度如何变化一直以来是基于性状的群落生态学的核心问题之一。尽管功能性状存在种内和种间变异, 但种内变异沿环境梯度如何变化仍有待探究。本文以鼎湖山南亚热带常绿阔叶林1.44 ha塔吊样地内16个树种的2,820个个体为研究对象, 探究4种叶功能性状(比叶面积、叶干物质含量、叶厚度和叶面积)沿群落垂直层次的种内变异。首先, 利用随机效应线性模型量化塔吊样地内的种内变异和种间变异; 其次, 利用Kmeans函数将森林的垂直层次划分为灌木层、亚冠层和林冠层, 并通过构建回归模型探究叶功能性状在群落垂直层次中的种内变异格局。最后, 应用混合线性模型和单因素方差分析的方法探究叶功能性状沿垂直层次的种内变异是否具有物种依赖性。结果表明: 在局域群落中, 并非所有叶功能性状的种内变异都低于种间变异; 叶功能性状在不同垂直层次的种内变异格局存在显著差异, 且种内变异与垂直范围呈正相关; 叶功能性状的种内变异具有较强的物种依赖性, 因此树种差异相对于小环境解释了更多的性状变异; 此外, 不同叶功能性状的种内变异沿垂直层次的变化趋势并不一致。本研究发现种内变异对于物种共存具有重要作用。     

Study of inheritance of feeding potential in natural populations of predatory coccinellid Cryptolaemus montrouzieri Mulsant using isofemale strains

The ability to feed on the prey is of great concern for the predatory insects, especially with regard to predatory coccinellid, Cryptolaemus montrouzieri Mulsant, which is mass reared and released into the field in large numbers to control the target pests. The variability associated with feeding potential is partly influenced by the genetic background of the insects and partly due to the environment, but the genetic basis of this trait is not yet fully understood in C. montrouzieri. The aim of this study was to identify the genetic basis of variation and heritability of this quantitative trait in natural populations of C. montrouzieri through isofemale heritability and parent–offspring regression. The regression analyses indicated that there was a significant linear relationship between progeny and their mothers for feeding potential.     

Regression-based quantitative trait loci mapping: robust, efficient and effective

Regression has always been an important tool for quantitative geneticists. The use of maximum likelihood (ML) has been advocated for the detection of quantitative trait loci (QTL) through linkage with molecular markers, and this approach can be very effective. However, linear regression models have also been proposed which perform similarly to ML, while retaining the many beneficial features of regression and, hence, can be more tractable and versatile than ML in some circumstances. Here, the use of linear regression to detect QTL in structured outbred populations is reviewed and its perceived shortfalls are revisited. It is argued that the approach is valuable now and will remain so in the future.     

The relationship between functional and taxonomic homogenization

Aim Species introductions and extinctions have reorganized the earth's biota, often leaving formerly spatially distinct assemblages more similar in species composition, a process termed biotic homogenization. The study of biotic homogenization has been almost entirely focused on the change in taxonomic similarity between assemblages through time. Here, we provide a trait‐based method for calculating functional similarity through time and compare these trends in functional attributes with those trends generated from a taxonomic perspective. Location Data were produced through computer simulation and gathered from North American Breeding Bird Survey (BBS) data and published accounts of North American birds for 10 locations across the east and west coast of the United States. Methods We simulated change in assemblages with different trait types (binary and continuous), levels of trait overlap, number of traits and species richness to determine the relationship between change in taxonomic similarity (ΔTS) and change in functional similarity (ΔFS). We also assess the relationship between ΔTS and ΔFS for bird assemblages across 10 locales in the USA between 1968 and 2008. We used simple linear regression to determine the slope and correlation between ΔTS and ΔFS and used multiple regression to assess the influence of trait overlap, number of traits, species richness and the ratio of traits to species on the relationship between ΔTS and ΔFS. Results Simulations reveal that trait redundancy governs the relationship between ΔTS and ΔFS. A decrease in trait overlap increases the slope of the regression between ΔTS and ΔFS and an increase in the ratio of traits to species in the regional pool increases the correlation. The relationship between ΔTS and ΔFS for breeding birds is comparable to simulations with low trait redundancy. Main conclusions We show that often losing or gaining species from an assemblage tells us very little about the loss or gain of function, and that this scenario most often occurs when the two assemblages have high trait redundancy. It remains to be seen how prevalent this scenario is within empirical examples; however, the implications for the continued delivery of ecosystem functions in the face of species introductions and extinctions are large.     

广义岭回归在家禽育种值估计中的应用

讨论了岭回归方法应用于混合线性模型方程组中估计家禽育种值的方法,其实质是将传统的混合线性模型方程组理解为一种广义岭回归估计,为确定遗传参数的估计提供了一种途径;同时,以番鸭为例,考虑了一个性状和两个固定效应,采用广义岭回归法对公番鸭育种值进行了估计,并与最佳线性无偏预测法(BLUP 法)进行了比较,结果表明,广义岭回归方法和BLUP 法估计的育种值及其排序非常接近,其相关系数和秩相关系数分别达到了0.998~(**)和0.986~(**),且采用广义岭回归法预测的误差率低(在±10%以内);表明在混合线性模型方程组中使用广义岭回归估计动物育种值的方法具有可行性,并可省去估计遗传参数的过程,使BLUP 法在动物选育中的应用更具实用性.     

Clustering expressed genes on the basis of their association with a quantitative phenotype

Cluster analyses of gene expression data are usually conducted based on their associations with the phenotype of a particular disease. Many disease traits have a clearly defined binary phenotype (presence or absence), so that genes can be clustered based on the differences of expression levels between the two contrasting phenotypic groups. For example, cluster analysis based on binary phenotype has been successfully used in tumour research. Some complex diseases have phenotypes that vary in a continuous manner and the method developed for a binary trait is not immediately applicable to a continuous trait. However, understanding the role of gene expression in these complex traits is of fundamental importance. Therefore, it is necessary to develop a new statistical method to cluster expressed genes based on their association with a quantitative trait phenotype. We developed a model-based clustering method to classify genes based on their association with a continuous phenotype. We used a linear model to describe the relationship between gene expression and the phenotypic value. The model effects of the linear model (linear regression coefficients) represent the strength of the association. We assumed that the model effects of each gene follow a mixture of several multivariate Gaussian distributions. Parameter estimation and cluster assignment were accomplished via an Expectation-Maximization (EM) algorithm. The method was verified by analysing two simulated datasets, and further demonstrated using real data generated in a microarray experiment for the study of gene expression associated with Alzheimer's disease.     

A unified Haseman-Elston method for testing linkage with quantitative traits

The Haseman and Elston (H-E) method uses a simple linear regression to model the squared trait difference of sib pairs with the shared allele identical by descent (IBD) at marker locus for linkage testing. Under this setting, the squared mean-corrected trait sum is also linearly related to the IBD sharing. However, the resulting slope estimate for either model is not efficient. In this report, we propose a simple linkage test that optimally uses information from the estimates of both models. We also demonstrate that the new test is more powerful than both the traditional one and the recently revisited H-E methods.     

Adjusting for confounding due to population admixture when estimating the effect of candidate genes on quantitative traits

When analyzing the relationship between allelic variability and traits, a potential source of confounding is population admixture. An approach to adjusting for potential confounding due to population admixture when estimating the influence of allelic variability at a candidate gene is presented. The approach involves augmenting linear regression models with additional regressors. Family genotype data are used to define the regressors, and inclusion of the regressors ensures that, even in the presence of population admixture, the estimates of the regression coefficients that parameterize the influence of allelic variability on the trait are unbiased. The approach is illustrated through an analysis of the influence of apolipoprotein E genotype on plasma low density lipoprotein cholesterol concentrations.     

Nonparametric evaluation of quantitative traits in population-based association studies when the genetic model is unknown

Statistical association between a single nucleotide polymorphism (SNP) genotype and a quantitative trait in genome-wide association studies is usually assessed using a linear regression model, or, in the case of non-normally distributed trait values, using the Kruskal-Wallis test. While linear regression models assume an additive mode of inheritance via equi-distant genotype scores, Kruskal-Wallis test merely tests global differences in trait values associated with the three genotype groups. Both approaches thus exhibit suboptimal power when the underlying inheritance mode is dominant or recessive. Furthermore, these tests do not perform well in the common situations when only a few trait values are available in a rare genotype category (disbalance), or when the values associated with the three genotype categories exhibit unequal variance (variance heterogeneity). We propose a maximum test based on Marcus-type multiple contrast test for relative effect sizes. This test allows model-specific testing of either dominant, additive or recessive mode of inheritance, and it is robust against variance heterogeneity. We show how to obtain mode-specific simultaneous confidence intervals for the relative effect sizes to aid in interpreting the biological relevance of the results. Further, we discuss the use of a related all-pairwise comparisons contrast test with range preserving confidence intervals as an alternative to Kruskal-Wallis heterogeneity test. We applied the proposed maximum test to the Bogalusa Heart Study dataset, and gained a remarkable increase in the power to detect association, particularly for rare genotypes. Our simulation study also demonstrated that the proposed non-parametric tests control family-wise error rate in the presence of non-normality and variance heterogeneity contrary to the standard parametric approaches. We provide a publicly available R library nparcomp that can be used to estimate simultaneous confidence intervals or compatible multiplicity-adjusted p-values associated with the proposed maximum test.     

Marker-assisted selection using ridge regression

In cross between inbred lines, linear regression can be used to estimate the correlation of markers with a trait of interest; these marker effects then allow marker assisted selection (MAS) for quantitative traits. Usually a subset of markers to include in the model must be selected: no completely satisfactory method of doing this exists. We show that replacing this selection of markers by ridge regression can improve the mean response to selection and reduce the variability of selection response.     

A genome scan for quantitative trait loci in a wild population of red deer (Cervus elaphus)

Recent empirical evidence indicates that although fitness and fitness components tend to have low heritability in natural populations, they may nonetheless have relatively large components of additive genetic variance. The molecular basis of additive genetic variation has been investigated in model organisms but never in the wild. In this article we describe an attempt to map quantitative trait loci (QTL) for birth weight (a trait positively associated with overall fitness) in an unmanipulated, wild population of red deer (Cervus elaphus). Two approaches were used: interval mapping by linear regression within half-sib families and a variance components analysis of a six-generation pedigree of >350 animals. Evidence for segregating QTL was found on three linkage groups, one of which was significant at the genome-wide suggestive linkage threshold. To our knowledge this is the first time that a QTL for any trait has been mapped in a wild mammal population. It is hoped that this study will stimulate further investigations of the genetic architecture of fitness traits in the wild.     

A comparison of restricted selection index and linear programming in sire selection

Summary The objective of restricted selection index is to enhance genetic change in one trait while restricting to zero change in a second trait. Linear programming is another, yet conceptually different, technique to maximize one function while enforcing limits on others. The objective of this research was to compare restricted selection index and linear programming in ability to maximize performance in one trait while limiting change in a second trait to zero. Results of a numerical study demonstrate that linear programming is a more effective method to limit correlated response than restricted selection index. On average, both methods limited response in a correlated trait to zero. However, the squared deviation of actual response in the restricted trait from zero was smaller with linear programming than with restricted selection index. Response to selection in the unrestricted trait is greater with restricted selection index than with linear programming.     

Inheritance and regression toward the mean in heterogeneous cell populations

Traits such as birth size and lifetime can vary widely even among non-mutated progeny of the same cell proliferating in the same environment. On the other hand, population parameters of these traits may remain stable over many generations, and there may be a distinct inheritance of these traits from mother to daughters. We have reconsidered the implication of mother-daughter correlations in light of linear regression analysis. It is proposed that a non-mutant cell whose phenotype deviates from the population mean produces progeny whose rate of regression toward the mean is proportional to 1-r, where r is the mother-daughter correlation coefficient of the trait under study. Theoretical support for this proposition is derived from linear regression analysis. Empirical support is found in pedigree analysis of cell growth constants among NIH3T3 mouse fibroblast cells, where the presence of an activated human ras oncogene is associated with a decreased r and an increased rate at which the growth constants of progeny regress toward the population mean.     

The spatial scale of local adaptation in a stochastic environment

The distribution of phenotypes in space will be a compromise between adaptive plasticity and local adaptation increasing the fit of phenotypes to local conditions and gene flow reducing that fit. Theoretical models on the evolution of quantitative characters on spatially explicit landscapes have only considered scenarios where optimum trait values change as deterministic functions of space. Here, these models are extended to include stochastic spatially autocorrelated aspects to the environment, and consequently the optimal phenotype. Under these conditions, the regression of phenotype on the environmental variable becomes steeper as the spatial scale on which populations are sampled becomes larger. Under certain deterministic models – such as linear clines – the regression is constant. The way in which the regression changes with spatial scale is informative about the degree of phenotypic plasticity, the relative scale of effective gene flow and the environmental dependency of selection. Connections to temporal models are discussed.     

Linear and generalized linear models for the detection of QTL effects on within-subject variability

Quantitative trait loci (QTLs) may affect not only the mean of a trait but also its variability. A special aspect is the variability between multiple measured traits of genotyped animals, such as the within-litter variance of piglet birth weights. The sample variance of repeated measurements is assigned as an observation for every genotyped individual. It is shown that the conditional distribution of the non-normally distributed trait can be approximated by a gamma distribution. To detect QTL effects in the daughter design, a generalized linear model with the identity link function is applied. Suitable test statistics are constructed to test the null hypothesis H(0): No QTL with effect on the within-litter variance is segregating versus H(A): There is a QTL with effect on the variability of birth weight within litter. Furthermore, estimates of the QTL effect and the QTL position are introduced and discussed. The efficiency of the presented tests is compared with a test based on weighted regression. The error probability of the first type as well as the power of QTL detection are discussed and compared for the different tests.