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.     

A simulation study on the accuracy of position and effect estimates of linked QTL and their asymptotic standard deviations using multiple interval mapping in an F2 scheme

Approaches like multiple interval mapping using a multiple-QTL model for simultaneously mapping QTL can aid the identification of multiple QTL, improve the precision of estimating QTL positions and effects, and are able to identify patterns and individual elements of QTL epistasis. Because of the statistical problems in analytically deriving the standard errors and the distributional form of the estimates and because the use of resampling techniques is not feasible for several linked QTL, there is the need to perform large-scale simulation studies in order to evaluate the accuracy of multiple interval mapping for linked QTL and to assess confidence intervals based on the standard statistical theory. From our simulation study it can be concluded that in comparison with a monogenetic background a reliable and accurate estimation of QTL positions and QTL effects of multiple QTL in a linkage group requires much more information from the data. The reduction of the marker interval size from 10 cM to 5 cM led to a higher power in QTL detection and to a remarkable improvement of the QTL position as well as the QTL effect estimates. This is different from the findings for (single) interval mapping. The empirical standard deviations of the genetic effect estimates were generally large and they were the largest for the epistatic effects. These of the dominance effects were larger than those of the additive effects. The asymptotic standard deviation of the position estimates was not a good criterion for the accuracy of the position estimates and confidence intervals based on the standard statistical theory had a clearly smaller empirical coverage probability as compared to the nominal probability. Furthermore the asymptotic standard deviation of the additive, dominance and epistatic effects did not reflect the empirical standard deviations of the estimates very well, when the relative QTL variance was smaller/equal to 0.5. The implications of the above findings are discussed.     

High-density genotyping: an overkill for QTL mapping? Lessons learned from a case study in maize and simulations

High-density genotyping is extensively exploited in genome-wide association mapping studies and genomic selection in maize. By contrast, linkage mapping studies were until now mostly based on low-density genetic maps and theoretical results suggested this to be sufficient. This raises the question, if an increase in marker density would be an overkill for linkage mapping in biparental populations, or if important QTL mapping parameters would benefit from it. In this study, we addressed this question using experimental data and a simulation based on linkage maps with marker densities of 1, 2, and 5 cM. QTL mapping was performed for six diverse traits in a biparental population with 204 doubled haploid maize lines and in a simulation study with varying QTL effects and closely linked QTL for different population sizes. Our results showed that high-density maps neither improved the QTL detection power nor the predictive power for the proportion of explained genotypic variance. By contrast, the precision of QTL localization, the precision of effect estimates of detected QTL, especially for small and medium sized QTL, as well as the power to resolve closely linked QTL profited from an increase in marker density from 5 to 1 cM. In conclusion, the higher costs for high-density genotyping are compensated for by more precise estimates of parameters relevant for knowledge-based breeding, thus making an increase in marker density for linkage mapping attractive.     

Application of a new IBD-based QTL mapping method to common wheat breeding population: analysis of kernel hardness and dough strength

Mapping quantitative trait loci (QTL) in plants is usually conducted using a population derived from a cross between two inbred lines. The power of such QTL detection and the estimation of the effects highly depend on the choice of the two parental lines. Thus, the QTL found represent only a small part of the genetic architecture and can be of limited economical interest in marker-assisted selection. On the other hand, applied breeding programmes evaluate large numbers of progeny derived from multiple-related crosses for a wide range of agronomic traits. It is assumed that the development of statistical techniques to deal with pedigrees in existing plant populations would increase the relevance and cost effectiveness of QTL mapping in a breeding context. In this study, we applied a two-step IBD-based-variance component method to a real wheat breeding population, composed of 374 F₆ lines derived from 80 different parents. Two bread wheat quality related traits were analysed by the method. Results obtained show very close agreement with major genes and QTL already known for those two traits. With this new QTL mapping strategy, inferences about QTL can be drawn across the breeding programme rather than being limited to the sample of progeny from a single cross and thus the use of the detected QTL in assisting breeding would be facilitated.     

Mapping quantitative trait loci in F2 incorporating phenotypes of F3 progeny

In plants and laboratory animals, QTL mapping is commonly performed using F(2) or BC individuals derived from the cross of two inbred lines. Typical QTL mapping statistics assume that each F(2) individual is genotyped for the markers and phenotyped for the trait. For plant traits with low heritability, it has been suggested to use the average phenotypic values of F(3) progeny derived from selfing F(2) plants in place of the F(2) phenotype itself. All F(3) progeny derived from the same F(2) plant belong to the same F(2:3) family, denoted by F(2:3). If the size of each F(2:3) family (the number of F(3) progeny) is sufficiently large, the average value of the family will represent the genotypic value of the F(2) plant, and thus the power of QTL mapping may be significantly increased. The strategy of using F(2) marker genotypes and F(3) average phenotypes for QTL mapping in plants is quite similar to the daughter design of QTL mapping in dairy cattle. We study the fundamental principle of the plant version of the daughter design and develop a new statistical method to map QTL under this F(2:3) strategy. We also propose to combine both the F(2) phenotypes and the F(2:3) average phenotypes to further increase the power of QTL mapping. The statistical method developed in this study differs from published ones in that the new method fully takes advantage of the mixture distribution for F(2:3) families of heterozygous F(2) plants. Incorporation of this new information has significantly increased the statistical power of QTL detection relative to the classical F(2) design, even if only a single F(3) progeny is collected from each F(2:3) family. The mixture model is developed on the basis of a single-QTL model and implemented via the EM algorithm. Substantial computer simulation was conducted to demonstrate the improved efficiency of the mixture model. Extension of the mixture model to multiple QTL analysis is developed using a Bayesian approach. The computer program performing the Bayesian analysis of the simulated data is available to users for real data analysis.     

Regression-based multi-trait QTL mapping using a structural equation model

Quantitative trait loci (QTL) mapping often results in data on a number of traits that have well-established causal relationships. Many multi-trait QTL mapping methods that account for the correlation among multiple traits have been developed to improve the statistical power and the precision of QTL parameter estimation. However, none of these methods are capable of incorporating the causal structure among the traits. Consequently, genetic functions of the QTL may not be fully understood. Structural equation modeling (SEM) allows researchers to explicitly characterize the causal structure among the variables and to decompose effects into direct, indirect, and total effects. In this paper, we developed a multi-trait SEM method of QTL mapping that takes into account the causal relationships among traits related to grain yield. Performance of the proposed method is evaluated by simulation study and applied to data from a wheat experiment. Compared with single trait analysis and the multi-trait least-squares analysis, our multi-trait SEM improves statistical power of QTL detection and provides important insight into how QTLs regulate traits by investigating the direct, indirect, and total QTL effects. The approach also helps build biological models that more realistically reflect the complex relationships among QTL and traits and is more precise and efficient in QTL mapping than single trait analysis.     

A model for quantitative trait loci mapping,linkage phase,and segregation pattern estimation for a full-sib progeny

Quantitative trait loci (QTL) mapping is an important approach for the study of the genetic architecture of quantitative traits. For perennial species, inbred lines cannot be obtained due to inbreed depression and a long juvenile period. Instead, linkage mapping can be performed by using a full-sib progeny. This creates a complex scenario because both markers and QTL alleles can have different segregation patterns as well as different linkage phases between them. We present a two-step method for QTL mapping using full-sib progeny based on composite interval mapping (i.e., interval mapping with cofactors), considering an integrated genetic map with markers with different segregation patterns and conditional probabilities obtained by a multipoint approach. The model is based on three orthogonal contrasts to estimate the additive effect (one in each parent) and dominance effect. These estimatives are obtained using the EM algorithm. In the first step, the genome is scanned to detect QTL. After, segregation pattern and linkage phases between QTL and markers are estimated. A simulated example is presented to validate the methodology. In general, the new model is more effective than existing approaches, because it can reveal QTL present in a full-sib progeny that segregates in any pattern present and can also identify dominance effects. Also, the inclusion of cofactors provided more statistical power for QTL mapping.     

Functional mapping of quantitative trait loci underlying growth trajectories using a transform-both-sides logistic model

The incorporation of developmental control mechanisms of growth has proven to be a powerful tool in mapping quantitative trait loci (QTL) underlying growth trajectories. A theoretical framework for implementing a QTL mapping strategy with growth laws has been established. This framework can be generalized to an arbitrary number of time points, where growth is measured, and becomes computationally more tractable, when the assumption of variance stationarity is made. In practice, however, this assumption is likely to be violated for age-specific growth traits due to a scale effect. In this article, we present a new statistical model for mapping growth QTL, which also addresses the problem of variance stationarity, by using a transform-both-sides (TBS) model advocated by Carroll and Ruppert (1984, Journal of the American Statistical Association 79, 321-328). The TBS-based model for mapping growth QTL cannot only maintain the original biological properties of a growth model, but also can increase the accuracy and precision of parameter estimation and the power to detect a QTL responsible for growth differentiation. Using the TBS-based model, we successfully map a QTL governing growth trajectories to a linkage group in an example of forest trees. The statistical and biological properties of the estimates of this growth QTL position and effect are investigated using Monte Carlo simulation studies. The implications of our model for understanding the genetic architecture of growth are discussed.     

Cross-validation in association mapping and its relevance for the estimation of QTL parameters of complex traits

Association mapping has become a widely applied genomic approach to identify quantitative trait loci (QTL) and dissect the genetic architecture of complex traits. However, approaches to assess the quality of the obtained QTL results are lacking. We therefore evaluated the potential of cross-validation in association mapping based on a large sugar beet data set. Our results show that the proportion of the population that should be used as estimation and validation sets, respectively, depends on the size of the mapping population. Generally, a fivefold cross-validation, that is, 20% of the lines as independent validation set, appears appropriate for commonly used population sizes. The predictive power for the proportion of genotypic variance explained by QTL was overestimated by on average 38% indicating a strong bias in the estimated QTL effects. The cross-validated predictive power ranged between 4 and 50%, which are more realistic estimates of this parameter for complex traits. In addition, QTL frequency distributions can be used to assess the precision of QTL position estimates and the robustness of the detected QTL. In summary, cross-validation can be a valuable tool to assess the quality of QTL parameters in association mapping.     

Use of randomization testing to detect multiple epistatic QTLs

Here, we describe a randomization testing strategy for mapping interacting quantitative trait loci (QTLs). In a forward selection strategy, non-interacting QTLs and simultaneously mapped interacting QTL pairs are added to a total genetic model. Simultaneous mapping of epistatic QTLs increases the power of the mapping strategy by allowing detection of interacting QTL pairs where none of the QTL can be detected by their marginal additive and dominance effects. Randomization testing is used to derive empirical significance thresholds for every model selection step in the procedure. A simulation study was used to evaluate the statistical properties of the proposed randomization tests and for which types of epistasis simultaneous mapping of epistatic QTLs adds power. Least squares regression was used for QTL parameter estimation but any other QTL mapping method can be used. A genetic algorithm was used to search for interacting QTL pairs, which makes the proposed strategy feasible for single processor computers. We believe that this method will facilitate the evaluation of the importance at epistatic interaction among QTLs controlling multifactorial traits and disorders.     

On the differences between maximum likelihood and regression interval mapping in the analysis of quantitative trait loci

The differences between maximum-likelihood (ML) and regression (REG) interval mapping in the analysis of quantitative trait loci (QTL) are investigated analytically and numerically by simulation. The analytical investigation is based on the comparison of the solution sets of the ML and REG methods in the estimation of QTL parameters. Their differences are found to relate to the similarity between the conditional posterior and conditional probabilities of QTL genotypes and depend on several factors, such as the proportion of variance explained by QTL, relative QTL position in an interval, interval size, difference between the sizes of QTL, epistasis, and linkage between QTL. The differences in mean squared error (MSE) of the estimates, likelihood-ratio test (LRT) statistics in testing parameters, and power of QTL detection between the two methods become larger as (1) the proportion of variance explained by QTL becomes higher, (2) the QTL locations are positioned toward the middle of intervals, (3) the QTL are located in wider marker intervals, (4) epistasis between QTL is stronger, (5) the difference between QTL effects becomes larger, and (6) the positions of QTL get closer in QTL mapping. The REG method is biased in the estimation of the proportion of variance explained by QTL, and it may have a serious problem in detecting closely linked QTL when compared to the ML method. In general, the differences between the two methods may be minor, but can be significant when QTL interact or are closely linked. The ML method tends to be more powerful and to give estimates with smaller MSEs and larger LRT statistics. This implies that ML interval mapping can be more accurate, precise, and powerful than REG interval mapping. The REG method is faster in computation, especially when the number of QTL considered in the model is large. Recognizing the factors affecting the differences between REG and ML interval mapping can help an efficient strategy, using both methods in QTL mapping to be outlined.     

A general statistical framework for mapping quantitative trait loci in nonmodel systems: issue for characterizing linkage phases

Because of uncertainty about linkage phases of founders, linkage mapping in nonmodel, outcrossing systems using molecular markers presents one of the major statistical challenges in genetic research. In this article, we devise a statistical method for mapping QTL affecting a complex trait by incorporating all possible QTL-marker linkage phases within a mapping framework. The advantage of this model is the simultaneous estimation of linkage phases and QTL location and effect parameters. These estimates are obtained through maximum-likelihood methods implemented with the EM algorithm. Extensive simulation studies are performed to investigate the statistical properties of our model. In a case study from a forest tree, this model has successfully identified a significant QTL affecting wood density. Also, the probability of the linkage phase between this QTL and its flanking markers is estimated. The implications of our model and its extension to more general circumstances are discussed.     

Quantitative trait loci x maternal cytoplasmic environment interaction for development rate in Oncorhynchus mykiss

Effects of maternal cytoplasmic environment (MCE) on development rate in rainbow trout were evaluated within a quantitative trait loci (QTL) analysis framework. Previous research had identified QTL for development rate in doubled haploid (DH) progeny produced from a cross between the Oregon State University (OSU) and the Swanson (SW) River rainbow trout clonal lines. In this study, progeny for QTL mapping were produced from a cross between the OSU and Clearwater (CW) River clonal lines. Doubled haploids were produced from the OSU x CW F1 by androgenesis using eggs from different females (or MCEs); with androgenesis, the maternal nuclear genome was destroyed by irradiation and diploidy was restored by blocking the first embryonic cleavage by heat shock. All embryos were incubated at the same temperature and development rate quantified as time to hatch. Using a linkage map constructed primarily with AFLP markers, QTL mapping was performed, including MCE covariates and QTL x MCE effects in models for testing. The major QTL for development rate in the OSU x SW cross overlaps with the major QTL found in this OSU x CW cross; effects at this locus were the same across MCEs. Both MCE and QTL x MCE effects contribute to variability in development rate, but QTL x MCE were minor and detected only at small-effect QTL.     

Bayesian analysis of genetic architecture of quantitative trait using data of crosses of multiple inbred lines

Using the data of crosses of multiple of inbred lines for mapping QTL can increase QTL detecting power compared with only cross of two inbred lines. Although many fixed-effect model methods have been proposed to analyze such data, they are largely based on one-QTL model or main effect model, and the interaction effects between QTL are always neglected. However, effectively separating the interaction effects from the residual error can increase the statistical power. In this article, we both extended the novel Bayesian model selection method and Bayesian shrinkage estimation approaches to multiple inbred line crosses. With two extensions, interacting QTL are effectively detected with high solution; in addition, the posterior variances for both main effects and interaction effects are also subjected to full Bayesian estimate, which is more optimal than two step approach involved in maximum-likelihood. A series of simulation experiments have been conducted to demonstrate the performance of the methods. The computer program written in FORTRAN language is freely available on request.     

Genetic mapping of allometric scaling laws

Many biological processes, from cellular metabolism to population dynamics, are characterized by particular allometric scaling relationships between rate and size (power laws). A statistical model for mapping specific quantitative trait loci (QTLs) that are responsible for allometric scaling laws has been developed. We present an improved model for allometric mapping of QTLs based on a more general allometry equation. This improved model includes two steps: (1) use model II regression analysis to estimate the parameters underlying universal allometric scaling laws, and (2) substitute the estimated allometric parameters in the mixture-based mapping model to obtain the estimation of QTL position and effects. This model has been validated by a real example for a mouse F2 progeny, in which two QTLs were detected on different chromosomes that determine the allometric relationship between growth rate and body weight.     

Using the mixed model for interval mapping of quantitative trait loci in outbred line crosses

Interval mapping by simple regression is a powerful method for the detection of quantitative trait loci (QTLs) in line crosses such as F2 populations. Due to the ease of computation of the regression approach, relatively complex models with multiple fixed effects, interactions between QTLs or between QTLs and fixed effects can easily be accommodated. However, polygenic effects, which are not targeted in QTL analysis, cannot be treated as random effects in a least squares analysis. In a cross between true inbred lines this is of no consequence, as the polygenic effect contributes just to the residual variance. In a cross between outbred lines, however, if a trait has high polygenic heritability, the additive polygenic effect has a large influence on variation in the population. Here we extend the fixed model for the regression interval mapping method to a mixed model using an animal model. This makes it possible to use not only the observations from progeny (e.g. F2), but also those from the parents (F1) to evaluate QTLs and polygenic effects. We show how the animal model using parental observations can be applied to an outbred cross and so increase the power and accuracy of QTL analysis. Three estimation methods, i.e. regression and an animal model either with or without parental observations, are applied to simulated data. The animal model using parental observations is shown to have advantages in estimating QTL position and additive genotypic value, especially when the polygenic heritability is large and the number of progeny per parent is small.     

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.     

Bayesian mapping of quantitative trait loci for complex binary traits

A complex binary trait is a character that has a dichotomous expression but with a polygenic genetic background. Mapping quantitative trait loci (QTL) for such traits is difficult because of the discrete nature and the reduced variation in the phenotypic distribution. Bayesian statistics are proved to be a powerful tool for solving complicated genetic problems, such as multiple QTL with nonadditive effects, and have been successfully applied to QTL mapping for continuous traits. In this study, we show that Bayesian statistics are particularly useful for mapping QTL for complex binary traits. We model the binary trait under the classical threshold model of quantitative genetics. The Bayesian mapping statistics are developed on the basis of the idea of data augmentation. This treatment allows an easy way to generate the value of a hypothetical underlying variable (called the liability) and a threshold, which in turn allow the use of existing Bayesian statistics. The reversible jump Markov chain Monte Carlo algorithm is used to simulate the posterior samples of all unknowns, including the number of QTL, the locations and effects of identified QTL, genotypes of each individual at both the QTL and markers, and eventually the liability of each individual. The Bayesian mapping ends with an estimation of the joint posterior distribution of the number of QTL and the locations and effects of the identified QTL. Utilities of the method are demonstrated using a simulated outbred full-sib family. A computer program written in FORTRAN language is freely available on request.