Comparison of statistical models to analyse the genetic effect on within-litter variance in pigs

Genetics affects not only the weight of piglets at birth but also the variability of birth weight within litter. Previous studies on this topic assigned the sample standard deviation of piglet birth weights within litter as an observation to the sow. However, the contribution of the difference in mean birth weight per sex on the within-litter variance has been neglected so far. This work deals with the genetic effect on within-litter variance when different statistical models with different distributional assumptions are used and considers the sex effect and appropriate weights per trait. Traits were formed from the pooled sample variance of male and female birth weights within litter. A linear model approach was fitted to the logarithmized within-litter variance and the sample standard deviation. A generalized linear model with gamma-distributed residuals and log-link function was applied to the untransformed sample variance. Models were compared by analysing data from 9439 litters from Landrace and Large White of a commercial breeding programme. The estimates of heritability for different traits ranged from 7% to 11%. Although the generalized linear mixed model is preferred from a mathematical view, the rank correlations between breeding values of the linear mixed models and the generalized linear mixed model were relatively high, i.e. 94% to 98%. By residual diagnostics, a linear mixed model using the weighted and pooled within-litter standard deviation was identified as most suitable.     

Power of different sampling strategies to detect quantitative trait loci variance effects

Summary Many studies have shown that segregating quantitative trait loci (QTL) can be detected via linkage to genetic markers. Power to detect a QTL effect on the trait mean as a function of the number of individuals genotyped for the marker is increased by selectively genotyping individuals with extreme values for the quantitative trait. Computer simulations were employed to study the effect of various sampling strategies on the statistical power to detect QTL variance effects. If only individuals with extreme phenotypes for the quantitative trait are selected for genotyping, then power to detect a variance effect is less than by random sampling. If 0.2 of the total number of individuals genotyped are selected from the center of the distribution, then power to detect a variance effect is equal to that obtained with random selection. Power to detect a variance effect was maximum when 0.2 to 0.5 of the individuals selected for genotyping were selected from the tails of the distribution and the remainder from the center.     

Genetic variability at neutral markers,quantitative trait land trait in a subdivided population under selection

Genetic variability in a subdivided population under stabilizing and diversifying selection was investigated at three levels: neutral markers, QTL coding for a trait, and the trait itself. A quantitative model with additive effects was used to link genotypes to phenotypes. No physical linkage was introduced. Using an analytical approach, we compared the diversity within deme (H(S)) and the differentiation (F(ST)) at the QTL with the genetic variance within deme (V(W)) and the differentiation (Q(ST)) for the trait. The difference between F(ST) and Q(ST) was shown to depend on the relative amounts of covariance between QTL within and between demes. Simulations were used to study the effect of selection intensity, variance of optima among demes, and migration rate for an allogamous and predominantly selfing species. Contrasting dynamics of the genetic variability at markers, QTL, and trait were observed as a function of the level of gene flow and diversifying selection. The highest discrepancy among the three levels occurred under highly diversifying selection and high gene flow. Furthermore, diversifying selection might cause substantial heterogeneity among QTL, only a few of them showing allelic differentiation, while the others behave as neutral markers.     

Identification of quantitative trait loci affecting cattle temperament

In addition to its potential contribution to improving animal welfare, the study of the genetics of cattle behavior may provide more general insights into the genetic control of such complex traits. We carried out a genome scan in a Holstein x Charolais cross cattle population to identify quantitative trait loci (QTL) influencing temperament-related traits. Individuals belonging to the second-generation of this population (F(2) and backcross individuals) were subjected to 2 behavioral tests. The flight from feeder (FF) test measured the distance at which the animal moved away from an approaching human observer, whereas the social separation (SS) test categorized different activities which the animal engaged in when removed from its penmates. The entire population was genotyped with 165 microsatellite markers. A regression interval mapping analysis identified 29 regions exceeding the 5% chromosome-wide significance level, which individually explained a relatively small fraction of the phenotypic variance of the traits (from 3.8% to 8.4%). One of the significant associations influencing an FF test trait on chromosome 29 reached the 5% genome-wide significance level. Eight other QTL, all associated with an SS test trait, reached the 1% chromosome-wide significance level. The location of some QTL coincided with other previously reported temperament QTL in cattle, whereas those that are reported for the first time here may represent general loci controlling temperament differences between cattle breeds. No overlapping QTL were identified for the traits measured by the 2 different tests, supporting the hypothesis that different genetic factors influence behavioral responses to different situations.     

Mapping quantitative trait Loci underlying fitness-related traits in a free-living sheep population

We searched for quantitative trait loci (QTL) underlying fitness-related traits in a free-living pedigree of 588 Soay sheep in which a genetic map using 251 markers with an average spacing of 15 cM had been established previously. Traits examined included birth date and weight, considered both as maternal and offspring traits, foreleg length, hindleg length, and body weight measured on animals in August and jaw length and metacarpal length measured on cleaned skeletal material. In some cases the data were split to consider different age classes separately, yielding a total of 15 traits studied. Genetic and environmental components of phenotypic variance were estimated for each trait and, for those traits showing nonzero heritability (N= 12), a QTL search was conducted by comparing a polygenic model with a model including a putative QTL. Support for a QTL at genome-wide significance was found on chromosome 11 for jaw length; suggestive QTL were found on chromosomes 2 and 5 (for birth date as a trait of the lamb), 8 (birth weight as a trait of the lamb), and 15 (adult hindleg length). We discuss the prospects for refining estimates of QTL position and effect size in the study population, and for QTL searches in free-living pedigrees in general.     

Bayesian analysis of linkage between genetic markers and quantitative trait loci. I. Prior knowledge

Summary Prior information on gene effects at individual quantitative trait loci (QTL) and on recombination rates between marker loci and QTL is derived. The prior distribution of QTL gene effects is assumed to be exponential with major effects less likely than minor ones. The prior probability of linkage between a marker and another single locus is a function of the number and length of chromosomes, and of the map function relating recombination rate to genetic distance among loci. The prior probability of linkage between a marker locus and a quantitative trait depends additionally on the number of detectable QTL, which may be determined from total additive genetic variance and minimum detectable QTL effect. The use of this prior information should improve linkage tests and estimates of QTL effects.     

Estimating genetic variance contributed by a quantitative trait locus: A random model approach

Detecting quantitative trait loci (QTL) and estimating QTL variances (represented by the squared QTL effects) are two main goals of QTL mapping and genome-wide association studies (GWAS). However, there are issues associated with estimated QTL variances and such issues have not attracted much attention from the QTL mapping community. Estimated QTL variances are usually biased upwards due to estimation being associated with significance tests. The phenomenon is called the Beavis effect. However, estimated variances of QTL without significance tests can also be biased upwards, which cannot be explained by the Beavis effect; rather, this bias is due to the fact that QTL variances are often estimated as the squares of the estimated QTL effects. The parameters are the QTL effects and the estimated QTL variances are obtained by squaring the estimated QTL effects. This square transformation failed to incorporate the errors of estimated QTL effects into the transformation. The consequence is biases in estimated QTL variances. To correct the biases, we can either reformulate the QTL model by treating the QTL effect as random and directly estimate the QTL variance (as a variance component) or adjust the bias by taking into account the error of the estimated QTL effect. A moment method of estimation has been proposed to correct the bias. The method has been validated via Monte Carlo simulation studies. The method has been applied to QTL mapping for the 10-week-body-weight trait from an F₂ mouse population.     

Estimation of genetic variance contributed by a quantitative trait locus: correcting the bias associated with significance tests

The Beavis effect in quantitative trait locus (QTL) mapping describes a phenomenon that the estimated effect size of a statistically significant QTL (measured by the QTL variance) is greater than the true effect size of the QTL if the sample size is not sufficiently large. This is a typical example of the Winners’ curse applied to molecular quantitative genetics. Theoretical evaluation and correction for the Winners’ curse have been studied for interval mapping. However, similar technologies have not been available for current models of QTL mapping and genome-wide association studies where a polygene is often included in the linear mixed models to control the genetic background effect. In this study, we developed the theory of the Beavis effect in a linear mixed model using a truncated noncentral Chi-square distribution. We equated the observed Wald test statistic of a significant QTL to the expectation of a truncated noncentral Chi-square distribution to obtain a bias-corrected estimate of the QTL variance. The results are validated from replicated Monte Carlo simulation experiments. We applied the new method to the grain width (GW) trait of a rice population consisting of 524 homozygous varieties with over 300 k single nucleotide polymorphism markers. Two loci were identified and the estimated QTL heritability were corrected for the Beavis effect. Bias correction for the larger QTL on chromosome 5 (GW5) with an estimated heritability of 12% did not change the QTL heritability due to the extremely large test score and estimated QTL effect. The smaller QTL on chromosome 9 (GW9) had an estimated QTL heritability of 9% reduced to 6% after the bias-correction.     

大白×梅山猪资源家系生长性状QTL的检测

以大白猪和梅山猪为父母本建立F2 资源家系 ,在 2 0 0 0年 ,随机选留 6 6头F2 代个体 ,获得出生重、6 0日龄体重、出生至 6 0日龄平均日增重及 6 0日龄至屠宰前平均日增重的表型数据。结合 4 8个微卫星标记构建的猪 1、2、3、4、6和 7号染色体遗传连锁图谱 ,用线性模型最小二乘法对各数量性状进行QTL区间作图 ,利用置换法(permutation)确定显著性阈值。研究发现 ,猪 4号染色体上有一个染色体水平极显著 (P <0 0 1)的QTL影响 6 0日龄至屠宰前平均日增重 ,并达到基因组显著水平 (P <0 0 5 )。在染色体水平 ,出生至 6 0日龄平均日增重QTL位于 2号染色体 ,6 0日龄体重QTL位于 1号染色体。 6号染色体的出生至 6 0日龄平均日增重QTL达到建议显著水平     

Contribution of an additive locus to genetic variance when inheritance is multi-factorial with implications on interpretation of GWAS

Although the effects of linkage disequilibrium (LD) on partition of genetic variance have received attention in quantitative genetics, there has been little discussion on how this phenomenon affects attribution of variance to a given locus. This paper reinforces the point that standard metrics used for assessing the contribution of a locus to variance can be misleading when there is linkage LD and that factors such as distribution of effects and of allelic frequencies over loci, or existence of frequency-dependent effects, play a role as well. An apparently new metric is proposed for measuring how much of the variability is contributed by a locus when LD exists. Effects of intervening factors, such as type and extent of LD, number of loci, distribution of effects, and of allelic frequencies over loci, as well as a model for generating frequency-dependent effects, are illustrated via hypothetical simulation scenarios. Implications on the interpretation of genome-wide association studies (GWAS), as typically carried out in human genetics, where single marker regression and the assumption of a sole quantitative trait locus (QTL) are common, are discussed. It is concluded that the standard attributions to variance contributed by a single QTL from a GWAS analysis may be misleading, conceptually and statistically, when a trait is complex and affected by sets of many genes in linkage disequilibrium. Yet another factor to consider in the “missing heritability” saga?.     

Genome‐scan analysis for genetic mapping of quantitative trait loci underlying birth weight and onset of puberty in doe kids (Capra hircus)

The objective of this study was to locate quantitative trait loci (QTL) causing variation in birth weight and age of puberty of doe kids in a population of Rayini cashmere goats. Four hundred and thirty kids from five half‐sib families were genotyped for 116 microsatellite markers located on the caprine autosomes. The traits recorded were birth weight of the male and female kids, body weight at puberty, average daily gain from birth to age of puberty and age at puberty of the doe kids. QTL analysis was conducted using the least squares interval mapping approach. Linkage analysis indicated significant QTL for birth weight on Capra hircus chromosomes (CHI) 4, 5, 6, 18 and 21. Five QTL located on CHI 5, 14 and 29 were associated with age at puberty. Across‐family analysis revealed evidence for overlapping QTL affecting birth weight (78 cM), body weight at puberty (72 cM), average daily gain from birth to age of puberty (72 cM) and age at puberty (76 cM) on CHI 5 and overlapping QTL controlling body weight at puberty and age at puberty on CHI 14 at 18–19 cM. The proportion of the phenotypic variance explained by the detected QTL ranged between 7.9% and 14.4%. Confirming some of the previously reported results for birth weight and growth QTL in goats, this study identified more QTL for these traits and is the first report of QTL for onset of puberty in doe kids.     

QTL detection with bidirectional and unidirectional selective genotyping: marker-based and trait-based analyses

Selective genotyping of one or both phenotypic extremes of a population can be used to detect linkage between markers and quantitative trait loci (QTL) in situations in which full-population genotyping is too costly or not feasible, or where the objective is to rapidly screen large numbers of potential donors for useful alleles with large effects. Data may be subjected to 'trait-based' analysis, in which marker allele frequencies are compared between classes of progeny defined based on trait values, or to 'marker-based' analysis, in which trait means are compared between progeny classes defined based on marker genotypes. Here, bidirectional and unidirectional selective genotyping were simulated, using population sizes and selection intensities relevant to cereal breeding. Control of Type I error was usually adequate with marker-based analysis of variance or trait-based testing using the normal approximation of the binomial distribution. Bidirectional selective genotyping was more powerful than unidirectional. Trait-based analysis and marker-based analysis of variance were about equally powerful. With genotyping of the best 30 out of 500 lines (6%), a QTL explaining 15% of the phenotypic variance could be detected with a power of 0.8 when tests were conducted at a marker 10 cM from the QTL. With bidirectional selective genotyping, QTL with smaller effects and (or) QTL farther from the nearest marker could be detected. Similar QTL detection approaches were applied to data from a population of 436 recombinant inbred rice lines segregating for a large-effect QTL affecting grain yield under drought stress. That QTL was reliably detected by genotyping as few as 20 selected lines (4.5%). In experimental populations, selective genotyping can reduce costs of QTL detection, allowing larger numbers of potential donors to be screened for useful alleles with effects across different backgrounds. In plant breeding programs, selective genotyping can make it possible to detect QTL using even a limited number of progeny that have been retained after selection.     

Genetic analysis and QTL mapping of the seed hardness trait in a black common bean (<Emphasis Type="Italic">Phaseolus vulgaris</Emphasis>) recombinant inbred line (RIL) population

Seed hardness trait has a profound impact on cooking time and canning quality in dry beans. This study aims to identify the unknown genetic factors and associated molecular markers to better understand and tag this trait. An F_2:7 recombinant inbred line (RIL) population was derived from a cross between the hard and soft seeded black bean parents (H68-4 and BK04-001). Eighty-five RILs and the parental lines were grown at two locations in southern Manitoba during years 2014–2016. Seed samples were harvested manually at maturity to test for seed hardness traits. The hydration capacity and stone seed count were estimated by soaking the seeds overnight at room temperature following AACC method 56-35.01. Seed samples from 2016 tests were also cooked to determine effect of seed hardness on cooking quality. For mapping of genomic regions contributing to the traits, the RIL population was genotyped using the genotype by sequencing (GBS) approach. The QTL mapping revealed that in addition to the major QTL on chromosome 7 at a genomic location previously reported to affect seed-hydration, two novel QTL with significant effects were also detected on chromosomes 1 and 2. In addition, a major QTL affecting the visual appeal of cooked bean was mapped on chromosome 4. This multi-year-site study shows that despite large environmental effects, seed hardness is an oligo-genic and highly heritable trait, which is inherited independently of the cooking quality scored as visual appeal of cooked beans. The identification of the QTLs and development of SNP markers associated with seed hardness can be applied for common bean variety improvement and genetic exploitation of these traits.     

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.     

Robust score statistics for QTL linkage analysis

The traditional variance components approach for quantitative trait locus (QTL) linkage analysis is sensitive to violations of normality and fails for selected sampling schemes. Recently, a number of new methods have been developed for QTL mapping in humans. Most of the new methods are based on score statistics or regression-based statistics and are expected to be relatively robust to non-normality of the trait distribution and also to selected sampling, at least in terms of type I error. Whereas the theoretical development of these statistics is more or less complete, some practical issues concerning their implementation still need to be addressed. Here we study some of these issues such as the choice of denominator variance estimates, weighting of pedigrees, effect of parameter misspecification, effect of non-normality of the trait distribution, and effect of incorporating dominance. We present a comprehensive discussion of the theoretical properties of various denominator variance estimates and of the weighting issue and then perform simulation studies for nuclear families to compare the methods in terms of power and robustness. Based on our analytical and simulation results, we provide general guidelines regarding the choice of appropriate QTL mapping statistics in practical situations.     

Modelling QTL effect on BTA06 using random regression test day models

In statistical models, a quantitative trait locus (QTL) effect has been incorporated either as a fixed or as a random term, but, up to now, it has been mainly considered as a time-independent variable. However, for traits recorded repeatedly, it is very interesting to investigate the variation of QTL over time. The major goal of this study was to estimate the position and effect of QTL for milk, fat, protein yields and for somatic cell score based on test day records, while testing whether the effects are constant or variable throughout lactation. The analysed data consisted of 23 paternal half-sib families (716 daughters of 23 sires) of Chinese Holstein-Friesian cattle genotyped at 14 microsatellites located in the area of the casein loci on BTA6. A sequence of three models was used: (i) a lactation model, (ii) a random regression model with a QTL constant in time and (iii) a random regression model with a QTL variable in time. The results showed that, for each production trait, at least one significant QTL exists. For milk and protein yields, the QTL effect was variable in time, while for fat yield, each of the three models resulted in a significant QTL effect. When a QTL is incorporated into a model as a constant over time, its effect is averaged over lactation stages and may, thereby, be difficult or even impossible to be detected. Our results showed that, in such a situation, only a longitudinal model is able to identify loci significantly influencing trait variation.     

Whole-genome scan for quantitative trait loci associated with birth weight, gestation length and passive immune transfer in a Holstein × Jersey crossbred population

We herein report results from a daughter design genome-scan study aiming to identify quantitative trait loci (QTL) associated with birth weight, direct gestation length and passive immune transfer in a backcross (Holstein × Jersey) × Holstein population. Two-hundred and seventy-six calves, offspring of seven crossbred sires, were genotyped for 161 microsatellite markers distributed along the 29 bovine autosomes. The genome scan was performed through interval mapping using an animal model in order to identify QTL accounting for phenotypic differences between individual animals. Based on significant chi-squared values, we identified putative QTL on BTA7 and BTA14 for gestation length, on BTA2, BTA6 and BTA14 for birth weight and on BTA20 for passive immune transfer. In total, these QTL accounted for 12%, 18% and 1% of the phenotypic variance in gestation length, birth weight and passive immune transfer respectively. We also report results from a supplementary and independent influential grand-daughter Holstein family. In this family, findings on BTA7 and BTA14 for direct gestation length were in agreement with results in the crossbred population. Two other regions on BTA6 and BTA21 putatively underlying QTL for direct gestation length variability were discovered with this analysis.     

Significance thresholds for quantitative trait locus mapping under selective genotyping

In the case of selective genotyping, the usual permutation test to establish statistical significance for quantitative trait locus (QTL) mapping can give inappropriate significance thresholds, especially when the phenotype distribution is skewed. A stratified permutation test should be used, with phenotypes shuffled separately within the genotyped and ungenotyped individuals.     

QTL mapping under truncation selection in homozygous lines derived from biparental crosses

In plant breeding, a large number of progenies that will be discarded later in the breeding process must be phenotyped and marker genotyped for conducting QTL analysis. In many cases, phenotypic preselection of lines could be useful. However, in QTL analyses even moderate preselection can have a significant effect on the power of QTL detection and estimation of effects of the target traits. In this study, we provide exact formulas for quantifying the change of allele frequencies within marker classes, expectations of marker contrasts and the variance of the marker contrasts under truncation selection, for the general case of two QTL affecting the target trait and a correlated trait. We focused on homozygous lines derived at random from biparental crosses. The effects of linkage between the marker and the QTL under selection as well as the effect of selection on a correlated trait can be quantified with the given formulas. Theoretical results clearly show that depending on the magnitude of QTL effects, high selection intensities can lead to a dramatic reduction in power of QTL detection and that approximations based on the infinitesimal model deviate substantially from exact solutions. The presented formulas are valuable for choosing appropriate selection intensity when performing QTL mapping experiments on the data on phenotypically preselected traits and enable the calculation and bias correction of the effects of QTL under selection. Application of our theory to experimental data revealed that selection-induced bias of QTL effects can be successfully corrected.