Power analysis for quantitative trait locus mapping in populations derived by multiple backcrosses

 Populations derived by multiple backcrosses are potentially useful for quantitative trait locus (QTL) mapping studies. Comparisons of relative power to detect QTL using populations derived by multiple back-crosses are needed to make decisions when mapping projects are initiated. The objective of this study was to theoretically compare the power to detect QTL in populations derived by multiple backcrosses relative to mapping in a recombinant inbred population of equal size. Backcrossing results in a reduction in genetic variance with each generation and also results in an increasing frequency of the recurrent parent marker genotype. The relevant outcome for QTL mapping is a reduction in genetic variance to partition between marker genotype classes and increasing unbalance of the number of individuals contributing to the mean of the marker genotypes. Both of these factors lead to a decrease in the power to detect a QTL as the number of backcross generations increases. Experimental error was held constant with the populations compared. From a theoretical standpoint, backcross-derived populations offer few advantages for QTL detection. If, however, a backcrossing approach is the most efficient method to achieve a desired breeding objective and if QTL detection is an objective of equal or less importance, backcross-derived populations are a reasonable approach to QTL detection. Received: 4 August 1996 / Accepted: 4 April 1997     

Power to detect higher-order epistatic interactions in a metabolic pathway using a new mapping strategy

Epistatic interactions among quantitative trait loci (QTL) contribute substantially to the variation in complex traits. The main objectives of this study were to (i) compare three- vs. four-step genome scans to identify three-way epistatic interactions among QTL belonging to a metabolic pathway, (ii) investigate by computer simulations the power and proportion of false positives (PFP) for detecting three-way interactions among QTL in recombinant inbred line (RIL) populations derived from a nested mating design, and (iii) compare these estimates to those obtained for detecting three-way interactions among QTL in RIL populations derived from diallel and different partial diallel mating designs. The single-nucleotide polymorphism haplotype data of B73 and 25 diverse maize inbreds were used to simulate the production of various RIL populations. Compared to the three-step genome scan, the power to detect three-way interactions was higher with the four-step genome scan. Higher power to detect three-way interactions was observed for RILs derived from optimally allocated distance-based designs than from nested designs or diallel designs. The power and PFP to detect three-way interactions using a nested design with 5000 RILs were for both the 4-QTL and the 12-QTL scenario of a magnitude that seems promising for their identification.     

Mapping multiple QTL of different effects: comparison of a simple sequential testing strategy and multiple QTL mapping

The aim of this study was to explore, by computer simulation, the mapping of QTLs in a realistic but complex situation of many (linked) QTLs with different effects, and to compare two QTL mapping methods. A novel method to dissect genetic variation on multiple chromosomes using molecular markers in backcross and F₂ populations derived from inbred lines was suggested, and its properties tested using simulations. The rationale for this sequential testing method was to explicitly test for alternative genetic models. The method consists of a series of four basic statistical tests to decide whether variance was due to a single QTL, two QTLs, multiple QTLs, or polygenes, starting with a test to detect genetic variance associated with a particular chromosome. The method was able to distinguish between different QTL configurations, in that the probability to `detect' the correct model was high, varying from 0.75 to 1. For example, for a backcross population of 200 and an overall heritability of 50%, in 78% of replicates a polygenic model was detected when that was the underlying true model. To test the method for multiple chromosomes, QTLs were simulated on 10 chromosomes, following a geometric series of allele effects, assuming positive alleles were in coupling in the founder lines For these simulations, the sequential testing method was compared to the established Multiple QTL Mapping (MQM) method. For a backcross population of 400 individuals, power to detect genetic variance was low with both methods when the heritability was 0.40. For example, the power to detect genetic variation on a chromosome on which 6 QTLs explained 12.6% of the genetic variance, was less than 60% for both methods. For a large heritability (0.90), the power of MQM to detect genetic variance and to dissect QTL configurations was generally better, due to the simultaneous fitting of markers on all chromosomes. It is concluded that when testing different QTL configurations on a single chromosome using the sequential testing procedure, regions of other chromosomes which explain a significant amount of variation should be fitted in the model of analysis. This study reinforces the need for large experiments in plants and other species if the aim of a genome scan is to dissect quantitative genetic variation.     

Evaluation of multi-locus models for genome-wide association studies: a case study in sugar beet

Association mapping has become a widely applied genomic approach to dissect the genetic architecture of complex traits. A major issue for association mapping is the need to control for the confounding effects of population structure, which is commonly done by mixed models incorporating kinship information. In this case study, we employed experimental data from a large sugar beet population to evaluate multi-locus models for association mapping. As in linkage mapping, markers are selected as cofactors to control for population structure and genetic background variation. We compared different biometric models with regard to important quantitative trait locus (QTL) mapping parameters like the false-positive rate, the QTL detection power and the predictive power for the proportion of explained genotypic variance. Employing different approaches we show that the multi-locus model, that is, incorporating cofactors, outperforms the other models, including the mixed model used as a reference model. Thus, multi-locus models are an attractive alternative for association mapping to efficiently detect QTL for knowledge-based breeding.     

Statistical issues in the analysis of quantitative traits in combined crosses.

We consider some practical statistical issues in QTL analysis where several crosses originate in multiple inbred parents. Our results show that ignoring background polygenic variation in different crosses may lead to biased interval mapping estimates of QTL effects or loss of efficiency. Threshold and power approximations are derived by extending earlier results based on the Ornstein-Uhlenbeck diffusion process. The results are useful in the design and analysis of genome screen experiments. Several common designs are evaluated in terms of their power to detect QTL.     

Empirical bayes method for incorporating data from multiple genome scans

Individual genome scans tend to have low power and can produce markedly biased estimates of QTL effects. Further, the confidence interval for their location is often prohibitively large for subsequent fine mapping and positional cloning. Given that a large number of genome scans have been conducted, not to mention the large number of variables and subsets tested, it is difficult to confidently rule out type 1 error as an explanation for significant effects even when there is apparent replication in a separate data set. We adapted Empirical Bayes (EB) methods [1] to analyze data from multiple genome scans simultaneously and alleviate each of these problems while still allowing for different QTL population effects across studies. We investigated the effects of using the EB method to include data from background studies to update the results of a single study of interest via simulation and demonstrated that it has a stable confidence level over a wide range of parameters defining the background studies and increased the power to detect linkage, even when some of the background studies were null or had QTL effect at other markers. This EB method for incorporating data from multiple studies into genome scan analyses seems promising.     

Genome-wide association mapping of agronomic traits in sugar beet

Recent results indicate that association mapping in populations from applied plant breeding is a powerful tool to detect QTL which are of direct relevance for breeding. The focus of this study was to unravel the genetic architecture of six agronomic traits in sugar beet. To this end, we employed an association mapping approach, based on a very large population of 924 elite sugar beet lines from applied plant breeding, fingerprinted with 677 single nucleotide polymorphism (SNP) markers covering the entire genome. We show that in this population linkage disequilibrium decays within a short genetic distance and is sufficient for the detection of QTL with a large effect size. To increase the QTL detection power and the mapping resolution a much higher number of SNPs is required. We found that for QTL detection, the mixed model including only the kinship matrix performed best, even in the presence of a considerable population structure. In genome-wide scans, main effect QTL and epistatic QTL were detected for all six traits. Our full two-dimensional epistasis scan revealed that for complex traits there appear to be epistatic master regulators, loci which are involved in a large number of epistatic interactions throughout the genome.     

The use of a genetic algorithm for simultaneous mapping of multiple interacting quantitative trait loci

Here we describe a general method for improving computational efficiency in simultaneous mapping of multiple interacting quantitative trait loci (QTL). The method uses a genetic algorithm to search for QTL in the genome instead of an exhaustive enumerative ("step-by-step") search. It can be used together with any method of QTL mapping based on a genomic search, since it only provides a more efficient way to search the genome for QTL. The computational demand decreases by a factor of approximately 130 when using genetic algorithm-based mapping instead of an exhaustive enumerative search for two QTL in a genome size of 2000 cM using a resolution of 1 cM. The advantage of using a genetic algorithm increases further for larger genomes, higher resolutions, and searches for more QTL. We show that a genetic algorithm-based search has efficiency higher than or equal to a search method conditioned on previously identified QTL for all epistatic models tested and that this efficiency is comparable to that of an exhaustive search for multiple QTL. The genetic algorithm is thus a powerful and computationally tractable alternative to the exhaustive enumerative search for simultaneous mapping of multiple interacting QTL. The use of genetic algorithms for simultaneous mapping of more than two QTL and for determining empirical significance thresholds using permutation tests is also discussed.     

On the use of mathematically-derived traits in QTL mapping

Mathematically-derived traits from two or more component traits, either by addition, subtraction, multiplication, or division, have been frequently used in genetics and breeding. When used in quantitative trait locus (QTL) mapping, derived traits sometimes show discrepancy with QTL identified for the component traits. We used three QTL distributions and three genetic effects models, and an actual maize mapping population, to investigate the efficiency of using derived traits in QTL mapping, and to understand the genetic and biological basis of derived-only QTL, i.e., QTL identified for a derived trait but not for any component trait. Results indicated that the detection power of the four putative QTL was consistently greater than 90% for component traits in simulated populations, each consisting of 200 recombinant inbred lines. Lower detection power and higher false discovery rate (FDR) were observed when derived traits were used. In an actual maize population, simulations were designed based on the observed QTL distributions and effects. When derived traits were used, QTL detected for both component and derived traits had comparable power, but those detected for component traits but not for derived traits had low detection power. The FDR from subtraction and division in the maize population were higher than the FDR from addition and multiplication. The use of derived traits increased the gene number, caused higher-order gene interactions than observed in component traits, and possibly complicated the linkage relationship between QTL as well. The increased complexity of the genetic architecture with derived traits may be responsible for the reduced detection power and the increased FDR. Derived-only QTL identified in practical genetic populations can be explained either as minor QTL that are not significant in QTL mapping of component traits, or as false positives.     

Mapping quantitative trait loci controlling silking date in a diallel cross among four lines of maize

 We describe and apply an interval mapping method for quantitative trait locus (QTL) detection using F₃ and testcross progenies derived from F₂ populations obtained from a diallel cross among four elite lines of maize. Linear model-based procedures were used for the test and estimation of putative QTL effects together with genetic interactions including epistasis. We mapped QTL associated with silking date and explored their genetic effects. Ten QTL were detected, and these explained more than 40% of the phenotypic variance. Most of these QTL had consistent and stable effects among genetic backgrounds and did not show significant epistasis. QTL-by-environment interaction was important for four QTL and was essentially due to changes in magnitude of allelic effects. These results show the efficiency of our method in several genetic situations as well as the power of the diallel design in detecting QTL simultaneously over several populations. Received: 2 September 1996 / Accepted: 20 December 1996     

Detection of quantitative trait Loci influencing recombination using recombinant inbred lines

The genetic basis of variation in recombination in higher plants is polygenic and poorly understood, despite its theoretical and practical importance. Here a method of detecting quantitative trait loci (QTL) influencing recombination in recombinant inbred lines (RILs) is proposed that relies upon the fact that genotype data within RILs carry the signature of past recombination. Behavior of the segregational genetic variance in numbers of chromosomal crossovers (recombination) over generations is described for self-, full-sib-, and half-sib-generated RILs with no dominance in true crossovers. This genetic variance, which as a fraction of the total phenotypic variance contributes to the statistical power of the method, was asymptotically greatest with half sibbing, less with sibbing, and least with selfing. The statistical power to detect a recombination QTL declined with diminishing QTL effect, genome target size, and marker density. For reasonably tight marker linkage power was greater with less intense inbreeding for later generations and vice versa for early generations. Generational optima for segregation variance and statistical power were found, whose onset and narrowness varied with marker density and mating design, being more pronounced for looser marker linkage. Application of this method to a maize RIL population derived from inbred lines Mo17 and B73 and developed by selfing suggested two putative QTL (LOD > 2.4) affecting certain chromosomes, and using a canonical transformation another putative QTL was detected. However, permutation tests failed to support their presence (experimentwise alpha = 0.05). Other populations with more statistical power and chosen specifically for recombination QTL segregation would be more effective.     

Connected populations for detecting quantitative trait loci and testing for epistasis: an application in maize

Quantitative trait loci (QTL) detection experiments have often been restricted to large biallelic populations. Use of connected multiparental crosses has been proposed to increase the genetic variability addressed and to test for epistatic interactions between QTL and the genetic background. We present here the results of a QTL detection performed on six connected F₂ populations of 150 F_2:3 families each, derived from four maize inbreds and evaluated for three traits of agronomic interest. The QTL detection was carried out by composite interval mapping on each population separately, then on the global design either by taking into account the connections between populations or not. Epistatic interactions between loci and with the genetic background were tested. Taking into account the connections between populations increased the number of QTL detected and the accuracy of QTL position estimates. We detected many epistatic interactions, particularly for grain yield QTL (R ² increase of 9.6%). Use of connections for the QTL detection also allowed a global ranking of alleles at each QTL. Allelic relationships and epistasis both contribute to the lack of consistency for QTL positions observed among populations, in addition to the limited power of the tests. The potential benefit of assembling favorable alleles by marker-assisted selection are discussed.     

Association mapping in multiple segregating populations of sugar beet (<Emphasis Type="Italic">Beta vulgaris</Emphasis> L.)

Association mapping in multiple segregating populations (AMMSP) combines high power to detect QTL in genome-wide approaches of linkage mapping with high mapping resolution of association mapping. The main objectives of this study were to (1) examine the applicability of AMMSP in a plant breeding context based on segregating populations of various size of sugar beet (Beta vulgaris L.), (2) compare different biometric approaches for AMMSP, and (3) detect markers with significant main effect across locations for nine traits in sugar beet. We used 768 F _n (n = 2, 3, 4) sugar beet genotypes which were randomly derived from 19 crosses among diploid elite sugar beet clones. For all nine traits, the genotypic and genotype × location interaction variances were highly significant (P < 0.01). Using a one-step AMMSP approach, the total number of significant (P < 0.05) marker-phenotype associations was 44. The identification of genome regions associated with the traits under consideration indicated that not only segregating populations derived from crosses of parental genotypes in a systematic manner could be used for AMMSP but also populations routinely derived in plant breeding programs from multiple, related crosses. Furthermore, our results suggest that data sets, whose size does not permit analysis by the one-step AMMSP approach, might be analyzed using the two-step approach based on adjusted entry means for each location without losing too much power for detection of marker-phenotype associations.     

Mapping quantitative trait loci by an extension of the Haley-Knott regression method using estimating equations

The Haley-Knott (HK) regression method continues to be a popular approximation to standard interval mapping (IM) of quantitative trait loci (QTL) in experimental crosses. The HK method is favored for its dramatic reduction in computation time compared to the IM method, something that is particularly important in simultaneous searches for multiple interacting QTL. While the HK method often approximates the IM method well in estimating QTL effects and in power to detect QTL, it may perform poorly if, for example, there is strong epistasis between QTL or if QTL are linked. Also, it is well known that the estimation of the residual variance by the HK method is biased. Here, we present an extension of the HK method that uses estimating equations based on both means and variances. For normally distributed phenotypes this estimating equation (EE) method is more efficient than the HK method. Furthermore, computer simulations show that the EE method performs well for very different genetic models and data set structures, including nonnormal phenotype distributions, nonrandom missing data patterns, varying degrees of epistasis, and varying degrees of linkage between QTL. The EE method retains key qualities of the HK method such as computational speed and robustness against nonnormal phenotype distributions, while approximating the IM method better in terms of accuracy and precision of parameter estimates and power to detect QTL.     

Quantitative trait loci (QTL) detection in multicross inbred designs: recovering QTL identical-by-descent status information from marker data

Mapping quantitative trait loci in plants is usually conducted using a population derived from a cross between two inbred lines. The power of such QTL detection and the parameter estimates depend largely on the choice of the two parental lines. Thus, the QTL detected in such populations represent only a small part of the genetic architecture of the trait. In addition, the effects of only two alleles are characterized, which is of limited interest to the breeder, while common pedigree breeding material remains unexploited for QTL mapping. In this study, we extend QTL mapping methodology to a generalized framework, based on a two-step IBD variance component approach, applicable to any type of breeding population obtained from inbred parents. We then investigate with simulated data mimicking conventional breeding programs the influence of different estimates of the IBD values on the power of QTL detection. The proposed method would provide an alternative to the development of specifically designed recombinant populations, by utilizing the genetic variation actually managed by plant breeders. The use of these detected QTL in assisting breeding would thus be facilitated.     

Combining data from multiple inbred line crosses improves the power and resolution of quantitative trait loci mapping

Rodent inbred line crosses are widely used to map genetic loci associated with complex traits. This approach has proven to be powerful for detecting quantitative trait loci (QTL); however, the resolution of QTL locations, typically approximately 20 cM, means that hundreds of genes are implicated as potential candidates. We describe analytical methods based on linear models to combine information available in two or more inbred line crosses. Our strategy is motivated by the hypothesis that common inbred strains of the laboratory mouse are derived from a limited ancestral gene pool and thus QTL detected in multiple crosses are likely to represent shared ancestral polymorphisms. We demonstrate that the combined-cross analysis can improve the power to detect weak QTL, can narrow support intervals for QTL regions, and can be used to separate multiple QTL that colocalize by chance. Moreover, combined-cross analysis can establish the allelic states of a QTL among a set of parental lines, thus providing critical information for narrowing QTL regions by haplotype analysis.     

Quantitative trait loci mapping in F(2) crosses between outbred lines

We develop a mixed-model approach for QTL analysis in crosses between outbred lines that allows for QTL segregation within lines as well as for differences in mean QTL effects between lines. We also propose a method called "segment mapping" that is based in partitioning the genome in a series of segments. The expected change in mean according to percentage of breed origin, together with the genetic variance associated with each segment, is estimated using maximum likelihood. The method also allows the estimation of differences in additive variances between the parental lines. Completely fixed random and mixed models together with segment mapping are compared via simulation. The segment mapping and mixed-model behaviors are similar to those of classical methods, either the fixed or random models, under simple genetic models (a single QTL with alternative alleles fixed in each line), whereas they provide less biased estimates and have higher power than fixed or random models in more complex situations, i.e., when the QTL are segregating within the parental lines. The segment mapping approach is particularly useful to determining which chromosome regions are likely to contain QTL when these are linked.     

Using linkage genome scans to improve power of association in genome scans

Scanning the genome for association between markers and complex diseases typically requires testing hundreds of thousands of genetic polymorphisms. Testing such a large number of hypotheses exacerbates the trade-off between power to detect meaningful associations and the chance of making false discoveries. Even before the full genome is scanned, investigators often favor certain regions on the basis of the results of prior investigations, such as previous linkage scans. The remaining regions of the genome are investigated simultaneously because genotyping is relatively inexpensive compared with the cost of recruiting participants for a genetic study and because prior evidence is rarely sufficient to rule out these regions as harboring genes with variation of conferring liability (liability genes). However, the multiple testing inherent in broad genomic searches diminishes power to detect association, even for genes falling in regions of the genome favored a priori. Multiple testing problems of this nature are well suited for application of the false-discovery rate (FDR) principle, which can improve power. To enhance power further, a new FDR approach is proposed that involves weighting the hypotheses on the basis of prior data. We present a method for using linkage data to weight the association P values. Our investigations reveal that if the linkage study is informative, the procedure improves power considerably. Remarkably, the loss in power is small, even when the linkage study is uninformative. For a class of genetic models, we calculate the sample size required to obtain useful prior information from a linkage study. This inquiry reveals that, among genetic models that are seemingly equal in genetic information, some are much more promising than others for this mode of analysis.     

Genome-wide evaluation for quantitative trait loci under the variance component model

The identity-by-descent (IBD) based variance component analysis is an important method for mapping quantitative trait loci (QTL) in outbred populations. The interval-mapping approach and various modified versions of it may have limited use in evaluating the genetic variances of the entire genome because they require evaluation of multiple models and model selection. In this study, we developed a multiple variance component model for genome-wide evaluation using both the maximum likelihood (ML) method and the MCMC implemented Bayesian method. We placed one QTL in every few cM on the entire genome and estimated the QTL variances and positions simultaneously in a single model. Genomic regions that have no QTL usually showed no evidence of QTL while regions with large QTL always showed strong evidence of QTL. While the Bayesian method produced the optimal result, the ML method is computationally more efficient than the Bayesian method. Simulation experiments were conducted to demonstrate the efficacy of the new methods.     

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.