Confidence Intervals in Qtl Mapping by Bootstrapping

The determination of empirical confidence intervals for the location of quantitative trait loci (QTLs) was investigated using simulation. Empirical confidence intervals were calculated using a bootstrap resampling method for a backcross population derived from inbred lines. Sample sizes were either 200 or 500 individuals, and the QTL explained 1, 5, or 10% of the phenotypic variance. The method worked well in that the proportion of empirical confidence intervals that contained the simulated QTL was close to expectation. In general, the confidence intervals were slightly conservatively biased. Correlations between the test statistic and the width of the confidence interval were strongly negative, so that the stronger the evidence for a QTL segregating, the smaller the empirical confidence interval for its location. The size of the average confidence interval depended heavily on the population size and the effect of the QTL. Marker spacing had only a small effect on the average empirical confidence interval. The LOD drop-off method to calculate empirical support intervals gave confidence intervals that generally were too small, in particular if confidence intervals were calculated only for samples above a certain significance threshold. The bootstrap method is easy to implement and is useful in the analysis of experimental data.     

Improved confidence intervals in quantitative trait loci mapping by permutation bootstrapping

The nonparametric bootstrap approach is known to be suitable for calculating central confidence intervals for the locations of quantitative trait loci (QTL). However, the distribution of the bootstrap QTL position estimates along the chromosome is peaked at the positions of the markers and is not tailed equally. This results in conservativeness and large width of the confidence intervals. In this study three modified methods are proposed to calculate nonparametric bootstrap confidence intervals for QTL locations, which compute noncentral confidence intervals (uncorrected method I), correct for the impact of the markers (weighted method I), or both (weighted method II). Noncentral confidence intervals were computed with an analog of the highest posterior density method. The correction for the markers is based on the distribution of QTL estimates along the chromosome when the QTL is not linked with any marker, and it can be obtained with a permutation approach. In a simulation study the three methods were compared with the original bootstrap method. The results showed that it is useful, first, to compute noncentral confidence intervals and, second, to correct the bootstrap distribution of the QTL estimates for the impact of the markers. The weighted method II, combining these two properties, produced the shortest and less biased confidence intervals in a large number of simulated configurations.     

Accounting for variability in the use of permutation testing to detect quantitative trait loci

Locating quantitative trait loci (QTL), or genomic regions associated with known molecular markers, is of increasing interest in a wide variety of applications ranging from human genetics to agricultural genetics. The hope of locating QTL (or genes) affecting a quantitative trait is that it will lead to characterization and possible manipulations of these genes. However, the complexity of both statistical and genetic issues surrounding the location of these regions calls into question the asymptotic statistical results supplying the distribution of the test statistics employed. Coupled with the power of current-day computing, permutation theory was reintroduced for the purpose of estimating the distribution of any test statistic used to test for the location of QTL. Permutation techniques have offered an attractive alternative to significance measures based on asymptotic theory. The ideas of permutation testing are extended in this application to include confidence intervals for the thresholds and p-values estimated in permutation testing procedures. The confidence intervals developed account for the Monte Carlo error associated with practical applications of permutation testing and lead to an effective method of determining an efficient permutation sample size.     

Advances in Statistical Methods to Map Quantitative Trait Loci in Outbred Populations

Statistical methods to map quantitative trait loci (QTL) in outbred populations are reviewed, extensions and applications to human and plant genetic data are indicated, and areas for further research are identified. Simple and computationally inexpensive methods include (multiple) linear regression of phenotype on marker genotypes and regression of squared phenotypic differences among relative pairs on estimated proportions of identity-by-descent at a locus. These methods are less suited for genetic parameter estimation in outbred populations but allow the determination of test statistic distributions via simulation or data permutation; however, further inferences including confidence intervals of QTL location require the use of Monte Carlo or bootstrap sampling techniques. A method which is intermediate in computational requirements is residual maximum likelihood (REML) with a covariance matrix of random QTL effects conditional on information from multiple linked markers. Testing for the number of QTLs on a chromosome is difficult in a classical framework. The computationally most demanding methods are maximum likelihood and Bayesian analysis, which take account of the distribution of multilocus marker-QTL genotypes on a pedigree and permit investigators to fit different models of variation at the QTL. The Bayesian analysis includes the number of QTLs on a chromosome as an unknown.     

DNA marker mining of ILSTS035 microsatellite locus on chromosome 6 of Hanwoo cattle

We describe tests for detecting and locating quantitative trait loci (QTL) for traits in Hanwoo cattle. From results of a permutation test to detect QTL for marbling, we selected the microsatellite locus ILSTS035 on chromosome 6 for further analysis. K-means clustering analysis applied to five traits and nine DNA markers in ILSTS035 resulted in three cluster groups. Finally we employed the bootstrap test method to calculate confidence intervals using the resampling method to find major DNA markers. We conclude that the major markers of ILSTS035 locus on chromosome 6 of Hanwoo cattle are markers 235 bp and 266 bp.     

Empirical nonparametric bootstrap strategies in quantitative trait loci mapping: conditioning on the genetic model.

Several nonparametric bootstrap methods are tested to obtain better confidence intervals for the quantitative trait loci (QTL) positions, i.e., with minimal width and unbiased coverage probability. Two selective resampling schemes are proposed as a means of conditioning the bootstrap on the number of genetic factors in our model inferred from the original data. The selection is based on criteria related to the estimated number of genetic factors, and only the retained bootstrapped samples will contribute a value to the empirically estimated distribution of the QTL position estimate. These schemes are compared with a nonselective scheme across a range of simple configurations of one QTL on a one-chromosome genome. In particular, the effect of the chromosome length and the relative position of the QTL are examined for a given experimental power, which determines the confidence interval size. With the test protocol used, it appears that the selective resampling schemes are either unbiased or least biased when the QTL is situated near the middle of the chromosome. When the QTL is closer to one end, the likelihood curve of its position along the chromosome becomes truncated, and the nonselective scheme then performs better inasmuch as the percentage of estimated confidence intervals that actually contain the real QTL''s position is closer to expectation. The nonselective method, however, produces larger confidence intervals. Hence, we advocate use of the selective methods, regardless of the QTL position along the chromosome (to reduce confidence interval sizes), but we leave the problem open as to how the method should be altered to take into account the bias of the original estimate of the QTL''s position.     

Application of non-parametric bootstrap methods to estimate confidence intervals for QTL location in a beef cattle QTL experimental population

Empirical confidence intervals (CIs) for the estimated quantitative trait locus (QTL) location from selective and non-selective non-parametric bootstrap resampling methods were compared for a genome scan involving an Angus x Brahman reciprocal fullsib backcross population. Genetic maps, based on 357 microsatellite markers, were constructed for 29 chromosomes using CRI-MAP V2.4. Twelve growth, carcass composition and beef quality traits (n = 527-602) were analysed to detect QTLs utilizing (composite) interval mapping approaches. CIs were investigated for 28 likelihood ratio test statistic (LRT) profiles for the one QTL per chromosome model. The CIs from the non-selective bootstrap method were largest (87 7 cM average or 79-2% coverage of test chromosomes). The Selective II procedure produced the smallest CI size (42.3 cM average). However, CI sizes from the Selective II procedure were more variable than those produced by the two LOD drop method. CI ranges from the Selective II procedure were also asymmetrical (relative to the most likely QTL position) due to the bias caused by the tendency for the estimated QTL position to be at a marker position in the bootstrap samples and due to monotonicity and asymmetry of the LRT curve in the original sample.     

On the mapping of quantitative trait loci at marker and non-marker locations

Previous studies have noted that the estimated positions of a large proportion of mapped quantitative trait loci (QTLs) coincide with marker locations and have suggested that this indicates a bias in the mapping methodology. In this study we predict the expected proportion of QTLs with positions estimated to be at the location of a marker and further examine the problem using simulated data. The results show that the higher proportion of putative QTLs estimated to be at marker positions compared with non-marker positions is an expected consequence of the estimation methods. The study initially focused on a single interval with no QTLs and was extended to include multiple intervals and QTLs of large effect. Further, the study demonstrated that the larger proportion of estimated QTL positions at the location of markers was not unique to linear regression mapping. Maximum likelihood produced similar results, although the accumulation of positional estimates at outermost markers was reduced when regions outside the linkage group were also considered. The bias towards marker positions is greatest under the null hypothesis of no QTLs or when QTL effects are small. This study discusses the impact the findings could have on the calculation of thresholds and confidence intervals produced by bootstrap methods.     

Genome-wide scan for bovine twinning rate QTL using linkage disequilibrium

Twinning is a complex trait with negative impacts on health and reproduction, which cause economic loss in dairy production. Several twinning rate quantitative trait loci (QTL) have been detected in previous studies, but confidence intervals for QTL location are broad and many QTL are unreplicated. To identify genomic regions or genes associated with twinning rate, QTL analysis based on linkage combined with linkage disequilibrium (LLD) and individual marker associations was conducted across the genome using high-throughput single nucleotide polymorphism (SNP) genotypes. A total of 9919 SNP markers were genotyped with 200 sires and sons in 19 half-sib North American Holstein dairy cattle families. After SNPs were genotyped, informative markers were selected for genome-wide association tests and QTL searches. Evidence for twinning rate QTL was found throughout the genome. Thirteen markers significantly associated with twinning rate were detected on chromosomes 2, 5 and 14 ( P  < 2.3 × 10⁻⁵). Twenty-six regions on fourteen chromosomes were identified by LLD analysis at P  < 0.0007. Seven previously reported ovulation or twinning rate QTL were supported by results of single marker association or LLD analyses. Single marker association analysis and LLD mapping were complementary tools for the identification of putative QTL in this genome scan.     

New molecular tools to improve the efficiency of breeding for increased drought resistance

The advent of molecular markers (particularly RFLP- and PCR-derived) for use as probes for genomic DNA has revolutionized the genetic analysis of crop plants and provided not only geneticists, but also physiologists, agronomists and breeders with valuable new tools to identify traits of importance in improving resistance to abiotic stresses. For the breeder, a genetic map saturated with molecular markers allows selection for certain characters to be carried out much more efficiently and effectively than was possible previously. Two areas of molecular marker technology that are proving particularly useful in identifying traits of value for stress resistance and introducing them into improved varieties are in situ hybridization with fluorescent-labelled molecular probes and quantitative trait locus (QTL) analysis with either radioactively- or cold-labelled probes. Fluorescence in situ hybridization (FISH) takes out much of the cytological tedium previously associated with monitoring the introgression of chromosomes and DNA fragments from one species to another. Labelled DNA can be prepared that is specific to a particular species and used to visualize in chromosome preparations the presence of chromosomes or chromosomal fragments from that species amongst the recipient's chromosomes. This is being used to help transfer genes for drought resistance and salt tolerance from alien species into Graminaceous crops. DNA probes showing polymorphism between the donor and recipient species can also be used to monitor the incorporation of alien genes from chromosome addition lines into the recipient species. High density molecular maps allow the location of all major genes regulating the expression of a particular trait to be determined. Statistical methods have been developed to allow QTL for the trait to be identified. Not only does this allow the complexity of genetic control of any trait to be determined, but by comparing the extent to which confidence intervals of QTL for different traits overlap it is possible to examine the likelihood that traits are pleiotropically linked. Thus, the traits most likely to be important in determining yield under droughted conditions can be identified. Examples are given of traits that could be incorporated into breeding programmes to improve drought resistance using techniques of marker-assisted selection.     

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.     

Fine-mapping of quantitative trait loci in half-sib families using current recombinations

Two groups of methods are being developed to fine-map quantitative trait loci (QTLs): identity-by-descent methods or methods using historical recombinations, and genetic chromosome dissection methods or methods utilizing current recombinations. Here we propose two methods that fall into the second group: contrast mapping and substitution mapping. A QTL has previously been detected via linkage mapping in a half-sib design (granddaughter or daughter design), and sires (grandsires) likely to be heterozygous at the QTL have been identified. A sire (grandsire) and its recombinant offspring are then genotyped for a series of ordered markers spanning the initial marker interval. Offspring are grouped by paternal multi-marker haplotype with haplotypes differing in the location of the recombination event. In the contrast method, contrasts between the phenotypic averages of haplotypes or offspring groups are calculated which correspond to marker intervals within the original interval. The expected value of the contrast for the true QTL interval is always maximum, hence the interval with maximum observed contrast is assumed to contain the QTL. Alternative statistics for determining the interval most likely to contain a QTL are presented for contrast mapping, as well as a bootstrap estimation of the probability of having identified the correct interval. For an initial marker bracket of 20 cM and 10 additional equidistant markers, the probability of assigning the QTL to the correct 2 cM marker interval or to a combined 4 cM interval was calculated. For substitution effects of 0.093, 0.232, 0.464, 0.696 and 0.928 (in additive genetic SD), power values near 0.14, 0.26, 0.48, 0.67 and 0.80 (0.25, 0.53, 0.86, 0.97 and 0.99) are achieved for a family of 200 (1000) sons, respectively. In substitution mapping, QTL segregation status of recombinant sons must be determined using daughter genotyping. Combinations of two haplotypes with their segregation status are required to assign the QTL to an interval. Probabilities of correct QTL assignment were calculated assuming absence of the mutant QTL allele in dams of sons. For a 2 cM interval and a QTL at the midpoint of an interval, power near 0.95 (0.90) is reached when the number of recombinant sons is 70 (60), or total number of sons is 424 (363). For QTL positions away from the midpoint, power decreases but can be improved by combining marker intervals. For a QTL located halfway to the midpoint, and 182 sons in a family resulting in 30 recombinant sons, probability is 0.94 for assignment to either a 2 cM or a combined 4 cM interval. Effect of type I and type II errors in segregation status determination on power of QTL assignment was found to be small. Errors in segregation status due to QTL segregation in dams have an impact if the frequency of the mutant QTL allele is intermediate to high.     

A region on chicken chromosome 2 affects both egg white thinning and egg weight

We describe the results from genetic dissection of a QTL region on chicken chromosome 2, shown to affect egg weight and quality in an earlier genome scan of an F₂ intercross between two divergent egg layer lines. As the 90% confidence intervals for the detected QTL covered tens of centiMorgans, new analyses were needed. The datasets were reanalysed with denser marker intervals to characterise the QTL region. Analysis of a candidate gene from the original QTL region, vimentin, did not support its role in controlling egg white thinning. Even after reanalysis with additional seven markers in the QTL area, the 90% confidence intervals remained large or even increased, suggesting the presence of multiple linked QTL for the traits. A grid search fitting two QTL on chromosome 2 for each trait suggested that there are two distinct QTL areas affecting egg white thinning in both production periods and egg weight in the late production period. The results indicate possible pleiotropic effects of some of the QTL on egg quality and egg weight. However, it was not possible to make a distinction between close linkage versus pleiotropic effects.     

The genetic basis of zinc tolerance in the metallophyte Arabidopsis halleri ssp. halleri (Brassicaceae): an analysis of quantitative trait loci

The species Arabidopsis halleri, an emerging model for the study of heavy metal tolerance and accumulation in plants, has evolved a high level of constitutive zinc tolerance. Mapping of quantitative trait loci (QTL) was used to investigate the genetic architecture of zinc tolerance in this species. A first-generation backcross progeny of A. halleri ssp. halleri from a highly contaminated industrial site and its nontolerant relative A. lyrata ssp. petraea was produced and used for QTL mapping of zinc tolerance. A genetic map covering most of the A. halleri genome was constructed using 85 markers. Among these markers, 65 were anchored in A. thaliana and revealed high synteny with other Arabidopsis genomes. Three QTL of comparable magnitude on three different linkage groups were identified. At all QTL positions zinc tolerance was enhanced by A. halleri alleles, indicating directional selection for higher zinc tolerance in this species. The two-LOD support intervals associated with these QTL cover 24, 4, and 13 cM. The importance of each of these three regions is emphasized by their colocalization with HMA4, MTP1-A, and MTP1-B, respectively, three genes well known to be involved in metal homeostasis and tolerance in plants.     

An empirical Bayes method for updating inferences in analysis of quantitative trait loci using information from related genome scans

Individual genome scans for quantitative trait loci (QTL) mapping often suffer from low statistical power and imprecise estimates of QTL location and effect. This lack of precision yields large confidence intervals for QTL location, which are problematic for subsequent fine mapping and positional cloning. In prioritizing areas for follow-up after an initial genome scan and in evaluating the credibility of apparent linkage signals, investigators typically examine the results of other genome scans of the same phenotype and informally update their beliefs about which linkage signals in their scan most merit confidence and follow-up via a subjective-intuitive integration approach. A method that acknowledges the wisdom of this general paradigm but formally borrows information from other scans to increase confidence in objectivity would be a benefit. We developed an empirical Bayes analytic method to integrate information from multiple genome scans. The linkage statistic obtained from a single genome scan study is updated by incorporating statistics from other genome scans as prior information. This technique does not require that all studies have an identical marker map or a common estimated QTL effect. The updated linkage statistic can then be used for the estimation of QTL location and effect. We evaluate the performance of our method by using extensive simulations based on actual marker spacing and allele frequencies from available data. Results indicate that the empirical Bayes method can account for between-study heterogeneity, estimate the QTL location and effect more precisely, and provide narrower confidence intervals than results from any single individual study. We also compared the empirical Bayes method with a method originally developed for meta-analysis (a closely related but distinct purpose). In the face of marked heterogeneity among studies, the empirical Bayes method outperforms the comparator.     

High-resolution quantitative trait locus mapping for body weight in mice by recombinant progeny testing

A major obstacle to the positional cloning of quantitative trait loci (QTLs) lies in resolving genetic factors whose allelic effects are blurred by environmental and background genetic variation. We investigate a fine-mapping approach that combines the use of an interval-specific congenic strain with progeny testing of recombinants for markers flanking a QTL. We apply the approach to map a murine QTL with an approximately 20% effect on growth rate by progeny testing 39 recombinants in a 12 cM region of the X chromosome. We use a likelihood analysis in an attempt to maximize the information on QTL map location and effect. The major X-linked effect is mapped to an approximately 2 cM region flanked by markers about 5 cM apart, outside which LOD support for the QTL drops extremely steeply by about 80. Nearly unambiguous assignment of the QTL genotypic state is obtained for each recombinant. The resolution of individual recombinants in the region is therefore sufficiently high to facilitate the positional cloning of the locus, although progress has been hampered because the genomic region containing the QTL shows an exceptionally low level of polymorphism in comparison with recent studies.     

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.     

Mapping multiple Quantitative Trait Loci by Bayesian classification

We developed a classification approach to multiple quantitative trait loci (QTL) mapping built upon a Bayesian framework that incorporates the important prior information that most genotypic markers are not cotransmitted with a QTL or their QTL effects are negligible. The genetic effect of each marker is modeled using a three-component mixture prior with a class for markers having negligible effects and separate classes for markers having positive or negative effects on the trait. The posterior probability of a marker's classification provides a natural statistic for evaluating credibility of identified QTL. This approach performs well, especially with a large number of markers but a relatively small sample size. A heat map to visualize the results is proposed so as to allow investigators to be more or less conservative when identifying QTL. We validated the method using a well-characterized data set for barley heading values from the North American Barley Genome Mapping Project. Application of the method to a new data set revealed sex-specific QTL underlying differences in glucose-6-phosphate dehydrogenase enzyme activity between two Drosophila species. A simulation study demonstrated the power of this approach across levels of trait heritability and when marker data were sparse.     

An analytical formula to estimate confidence interval of QTL location with a saturated genetic map as a function of experimental design

Analytical formulae are derived for the confidence interval for location of a quantitative trait locus (QTL) using a saturated genetic map, as a function of the experimental design, the QTL allele substitution effect, and the number of individuals genotyped and phenotyped. The formulae are derived assuming evenly spaced recombination events, rather than the actual unevenly spaced distribution. The formulae are useful for determining desired sample size when designing a wide variety of QTL mapping experiments, and for evaluating a priori the potential of a given mapping population for defining the location of a QTL. The formulae do not take into account the finite number of recombination events in a given sample.     

A paternally imprinted QTL for mature body mass on mouse Chromosome 8

Body mass (BM) is a classic polygenic trait that has been extensively investigated to determine the underlying genetic architecture. Many previous studies looking at the genetic basis of variation in BM in murine animal models by quantitative trait loci (QTL) mapping have used crosses between two inbred lines. As a consequence it has not been possible to explore imprinting effects which have been shown to play an important role in the genetic basis of early growth with persistent effects throughout the growth curve. Here we use partially inbred mouse lines to identify QTL for mature BM by applying both Mendelian and Imprinting models. The analysis of an F₂ population (n ≈ 500) identified a number of QTL at 14, 16, and 18 weeks explaining in total 31.5%, 34.4%, and 30.5% of total phenotypic variation, respectively. On Chromosome 8 a QTL of large effect (14% of the total phenotypic variance at 14 weeks) was found to be explained by paternal imprinting. Although Chromosome 8 has not been previously associated with imprinting effects, features of candidate genes within the QTL confidence interval (CpG islands and direct clustered repeats) support the hypothesis that Insulin receptor substrate 2 may be associated with imprinting, but as yet is unidentified as being so.