Relaxed significance criteria for linkage analysis

Linkage analysis involves performing significance tests at many loci located throughout the genome. Traditional criteria for declaring a linkage statistically significant have been formulated with the goal of controlling the rate at which any single false positive occurs, called the genomewise error rate (GWER). As complex traits have become the focus of linkage analysis, it is increasingly common to expect that a number of loci are truly linked to the trait. This is especially true in mapping quantitative trait loci (QTL), where sometimes dozens of QTL may exist. Therefore, alternatives to the strict goal of preventing any single false positive have recently been explored, such as the false discovery rate (FDR) criterion. Here, we characterize some of the challenges that arise when defining relaxed significance criteria that allow for at least one false positive linkage to occur. In particular, we show that the FDR suffers from several problems when applied to linkage analysis of a single trait. We therefore conclude that the general applicability of FDR for declaring significant linkages in the analysis of a single trait is dubious. Instead, we propose a significance criterion that is more relaxed than the traditional GWER, but does not appear to suffer from the problems of the FDR. A generalized version of the GWER is proposed, called GWERk, that allows one to provide a more liberal balance between true positives and false positives at no additional cost in computation or assumptions.     

Gene-environment interactions in complex diseases: genetic models and methods for QTL mapping in multiple half-sib populations

An interval quantitative trait locus (QTL) mapping method for complex polygenic diseases (as binary traits) showing QTL by environment interactions (QEI) was developed for outbred populations on a within-family basis. The main objectives, within the above context, were to investigate selection of genetic models and to compare liability or generalized interval mapping (GIM) and linear regression interval mapping (RIM) methods. Two different genetic models were used: one with main QTL and QEI effects (QEI model) and the other with only a main QTL effect (QTL model). Over 30 types of binary disease data as well as six types of continuous data were simulated and analysed by RIM and GIM. Using table values for significance testing, results show that RIM had an increased false detection rate (FDR) for testing interactions which was attributable to scale effects on the binary scale. GIM did not suffer from a high FDR for testing interactions. The use of empirical thresholds, which effectively means higher thresholds for RIM for testing interactions, could repair this increased FDR for RIM, but such empirical thresholds would have to be derived for each case because the amount of FDR depends on the incidence on the binary scale. RIM still suffered from higher biases (15-100% over- or under-estimation of true values) and high standard errors in QTL variance and location estimates than GIM for QEI models. Hence GIM is recommended for disease QTL mapping with QEI. In the presence of QEI, the model including QEI has more power (20-80% increase) to detect the QTL when the average QTL effect is small (in a situation where the model with a main QTL only is not too powerful). Top-down model selection is proposed in which a full test for QEI is conducted first and then the model is subsequently simplified. Methods and results will be applicable to human, plant and animal QTL mapping experiments.     

Controlling the proportion of false positives in multiple dependent tests

Genome scan mapping experiments involve multiple tests of significance. Thus, controlling the error rate in such experiments is important. Simple extension of classical concepts results in attempts to control the genomewise error rate (GWER), i.e., the probability of even a single false positive among all tests. This results in very stringent comparisonwise error rates (CWER) and, consequently, low experimental power. We here present an approach based on controlling the proportion of false positives (PFP) among all positive test results. The CWER needed to attain a desired PFP level does not depend on the correlation among the tests or on the number of tests as in other approaches. To estimate the PFP it is necessary to estimate the proportion of true null hypotheses. Here we show how this can be estimated directly from experimental results. The PFP approach is similar to the false discovery rate (FDR) and positive false discovery rate (pFDR) approaches. For a fixed CWER, we have estimated PFP, FDR, pFDR, and GWER through simulation under a variety of models to illustrate practical and philosophical similarities and differences among the methods.     

Joint-multiple family linkage analysis predicts within-family variation better than single-family analysis of the maize nested association mapping population

Quantitative trait locus (QTL) mapping has been used to dissect the genetic architecture of complex traits and predict phenotypes for marker-assisted selection. Many QTL mapping studies in plants have been limited to one biparental family population. Joint analysis of multiple biparental families offers an alternative approach to QTL mapping with a wider scope of inference. Joint-multiple population analysis should have higher power to detect QTL shared among multiple families, but may have lower power to detect rare QTL. We compared prediction ability of single-family and joint-family QTL analysis methods with fivefold cross-validation for 6 diverse traits using the maize nested association mapping population, which comprises 25 biparental recombinant inbred families. Joint-family QTL analysis had higher mean prediction abilities than single-family QTL analysis for all traits at most significance thresholds, and was always better at more stringent significance thresholds. Most robust QTL (detected in >50% of data samples) were restricted to one family and were often not detected at high frequency by joint-family analysis, implying substantial genetic heterogeneity among families for complex traits in maize. The superior predictive ability of joint-family QTL models despite important genetic differences among families suggests that joint-family models capture sufficient smaller effect QTL that are shared across families to compensate for missing some rare large-effect QTL.     

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.     

Parallel computing in interval mapping of quantitative trait loci.

Linear regression analysis is considered the least computationally demanding method for mapping quantitative trait loci (QTL). However, simultaneous search for multiple QTL, the use of permutations to obtain empirical significance thresholds, and larger experimental studies significantly increase the computational demand. This report describes an easily implemented parallel algorithm, which significantly reduces the computing time in both QTL mapping and permutation testing. In the example provided, the analysis time was decreased to less than 15% of a single processor system by the use of 18 processors. We indicate how the efficiency of the analysis could be improved by distributing the computations more evenly to the processors and how other ways of distributing the data facilitate the use of more processors. The use of parallel computing in QTL mapping makes it possible to routinely use permutations to obtain empirical significance thresholds for multiple traits and multiple QTL models. It could also be of use to improve the computational efficiency of the more computationally demanding QTL analysis methods.     

Multiple quantitative trait loci mapping with cofactors and application of alternative variants of the false discovery rate in an enlarged granddaughter design

The experimental power of a granddaughter design to detect quantitative trait loci (QTL) in dairy cattle is often limited by the availability of progeny-tested sires, by the ignoring of already identified QTL in the statistical analysis, and by the application of stringent experimentwise significance levels. This study describes an experiment that addressed these points. A large granddaughter design was set up that included sires from two countries (Germany and France), resulting in almost 2000 sires. The animals were genotyped for markers on nine different chromosomes. The QTL analysis was done for six traits separately using a multimarker regression that included putative QTL on other chromosomes as cofactors in the model. Different variants of the false discovery rate (FDR) were applied. Two of them accounted for the proportion of truly null hypotheses, which were estimated to be 0.28 and 0.3, respectively, and were therefore tailored to the experiment. A total of 25 QTL could be mapped when cofactors were included in the model-7 more than without cofactors. Controlling the FDR at 0.05 revealed 31 QTL for the two FDR methods that accounted for the proportion of truly null hypotheses. The relatively high power of this study can be attributed to the size of the experiment, to the QTL analysis with cofactors, and to the application of an appropriate FDR.     

Genetic and Quantitative Trait Locus Analysis for Bio-Oil Compounds after Fast Pyrolysis in Maize Cobs

Fast pyrolysis has been identified as one of the biorenewable conversion platforms that could be a part of an alternative energy future, but it has not yet received the same attention as cellulosic ethanol in the analysis of genetic inheritance within potential feedstocks such as maize. Ten bio-oil compounds were measured via pyrolysis/gas chromatography-mass spectrometry (Py/GC-MS) in maize cobs. 184 recombinant inbred lines (RILs) of the intermated B73 x Mo17 (IBM) Syn4 population were analyzed in two environments, using 1339 markers, for quantitative trait locus (QTL) mapping. QTL mapping was performed using composite interval mapping with significance thresholds established by 1000 permutations at α = 0.05. 50 QTL were found in total across those ten traits with R² values ranging from 1.7 to 5.8%, indicating a complex quantitative inheritance of these traits.     

Multiple trait multiple interval mapping of quantitative trait loci from inbred line crosses

ABSTRACT: BACKGROUND: Although many experiments have measurements on multiple traits, most studies performed the analysis of mapping of quantitative trait loci (QTL) for each trait separately using single trait analysis. Single trait analysis does not take advantage of possible genetic and environmental correlations between traits. In this paper, we propose a novel statistical method for multiple trait multiple interval mapping (MTMIM) of QTL for inbred line crosses. We also develop a novel score-based method for estimating genome-wide significance level of putative QTL effects suitable for the MTMIM model. The MTMIM method is implemented in the freely available and widely used Windows QTL Cartographer software. RESULTS: Throughout the paper, we provide compelling empirical evidences that: (1) the score-based threshold maintains proper type I error rate and tends to keep false discovery rate within an acceptable level; (2) the MTMIM method can deliver better parameter estimates and power than single trait multiple interval mapping method; (3) an analysis of Drosophila dataset illustrates how the MTMIM method can better extract information from datasets with measurements in multiple traits. CONCLUSIONS: The MTMIM method represents a convenient statistical framework to test hypotheses of pleiotropic QTL versus closely linked nonpleiotropic QTL, QTL by environment interaction, and to estimate the total genotypic variance-covariance matrix between traits and to decompose it in terms of QTL-specific variance-covariance matrices, therefore, providing more details on the genetic architecture of complex traits.     

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.     

Quantitative trait loci for grain moisture at harvest and field grain drying rate in maize (Zea mays, L.)

Hybrids with low grain moisture (GM) at harvest are specially required in mid- to short-season environments. One of the most important factors determining this trait is field grain drying rate (FDR). To produce hybrids with low GM at harvest, inbred lines can be obtained through selection for either GM or FDR. Thus, a single-cross population (181 F _2:3-generation plants) of two divergent inbred lines was evaluated to locate QTL affecting GM at harvest and FDR as a starting point for marker assisted selection (MAS). Moisture measurements were made with a hand-held moisture meter. Detection of QTL was facilitated with interval mapping in one and two dimensions including an interaction term, and a genetic linkage map of 122 SSR loci covering 1,557.8 cM. The markers were arranged in ten linkage groups. QTL mapping was made for the mean trait performance of the F _2:3 population across years. Ten QTL and an interaction were associated with GM. These QTL accounted for 54.8 and 65.2% of the phenotypic and genotypic variation, respectively. Eight QTL and two interactions were associated with FDR accounting for 35.7 and 45.2% of the phenotypic and genotypic variation, respectively. Two regions were in common between traits. The interaction between QTL for GM at harvest had practical implications for MAS. We conclude that MAS per se will not be an efficient method for reducing GM at harvest and/or increasing FDR. A selection index including both molecular marker information and phenotypic values, each appropriately weighted, would be the best selection strategy.     

The Statistical Power of Inclusive Composite Interval Mapping in Detecting Digenic Epistasis Showing Common F2 Segregation Ratios

Epistasis is a commonly observed genetic phenomenon and an important source of variation of complex traits,which could maintain additive variance and therefore assure the long-term genetic gain in breeding.Inclusive composite interval mapping（ICIM） is able to identify epistatic quantitative trait loci（QTLs） no matter whether the two interacting QTLs have any additive effects.In this article,we conducted a simulation study to evaluate detection power and false discovery rate（FDR） of ICIM epistatic mapping,by considering F₂ and doubled haploid（DH） populations,different F₂ segregation ratios and population sizes.Results indicated that estimations of QTL locations and effects were unbiased,and the detection power of epistatic mapping was largely affected by population size,heritability of epistasis,and the amount and distribution of genetic effects.When the same likelihood of odd（LOD） threshold was used,detection power of QTL was higher in F₂ population than power in DH population;meanwhile FDR in F₂ was also higher than that in DH.The increase of marker density from 10 cM to 5 cM led to similar detection power but higher FDR.In simulated populations,ICIM achieved better mapping results than multiple interval mapping（MIM） in estimation of QTL positions and effect.At the end,we gave epistatic mapping results of ICIM in one actual population in rice（Oryza sativa L.）.     

Efficiency of genome-wide association studies in random cross populations

Genome-wide association studies (GWAS) with plant species have employed inbred lines panels. We evaluated the efficiency of GWAS in non-inbred and inbred populations and assessed factors affecting GWAS. Fifty samples of 800 individuals from populations with linkage disequilibrium were simulated. Individuals were genotyped for 10,000 single nucleotide polymorphisms (SNPs) and phenotyped for traits controlled by ten quantitative trait loci (QTLs) and 90 minor genes, assuming different degrees of dominance and broad sense heritabilities of 40 and 80%. The average SNP density was 0.1 centiMorgan (cM) and the QTL heritabilities ranged from 3.2 to 11.8%. The results for random cross populations evidenced that to increase the QTL detection power, the additive-dominance model must be fitted for traits controlled by dominance effects but must not be fitted for traits showing no dominance. The power of detection was maximized by increasing the sample size to 400 and the false discovery rate (FDR) to 5%. The average power of detection for the low, intermediate, and high heritability QTLs achieved 52.4, 87.0, and 100.0%, respectively. Assuming sample sizes of 400 and 800, the observed FDR was equal to or lower than the specified level of significance. The association mapping was highly precise, since at least 97% of the declared QTLs were detected by the SNP inside it (average bias of 0.4 cM). Besides controlling the FDR, relatedness (and identity by state) efficiently controls the number of significant associations outside the QTL interval (not all false positive associations). The analysis of the inbred random cross population provided essentially the same results as the non-inbred populations.     

Power of quantitative trait locus mapping for polygenic binary traits using generalized and regression interval mapping in multi-family half-sib designs

A generalized interval mapping (GIM) method to map quantitative trait loci (QTL) for binary polygenic traits in a multi-family half-sib design is developed based on threshold theory and implemented using a Newton-Raphson algorithm. Statistical power and bias of QTL mapping for binary traits by GIM is compared with linear regression interval mapping (RIM) using simulation. Data on 20 paternal half-sib families were simulated with two genetic markers that bracketed an additive QTL. Data simulated and analysed were: (1) data on the underlying normally distributed liability (NDL) scale, (2) binary data created by truncating NDL data based on three thresholds yielding data sets with three different incidences, and (3) NDL data with polygenic and QTL effects reduced by a proportion equal to the ratio of the heritabilities on the binary versus NDL scale (reduced-NDL). Binary data were simulated with and without systematic environmental (herd) effects in an unbalanced design. GIM and RIM gave similar power to detect the QTL and similar estimates of QTL location, effects and variances. Presence of fixed effects caused differences in bias between RIM and GIM, where GIM showed smaller bias which was affected less by incidence. The original NDL data had higher power and lower bias in QTL parameter estimates than binary and reduced-NDL data. RIM for reduced-NDL and binary data gave similar power and estimates of QTL parameters, indicating that the impact of the binary nature of data on QTL analysis is equivalent to its impact on heritability.     

A quantitative trait locus for live weight maps to bovine Chromosome 23

A multiple-marker mapping approach was used to search for quantitative trait loci (QTLs) affecting production, health, and fertility traits in Finnish Ayrshire dairy cattle. As part of a whole-genome scan, altogether 469 bulls were genotyped for six microsatellite loci in 12 families on Chromosome (Chr) 23. Both multiple-marker interval mapping with regression and maximum-likelihood methods were applied with a granddaughter design. Eighteen traits, belonging to 11 trait groups, were included in the analysis. One QTL exceeded experiment level and one QTL genome level significance thresholds. Across-families analysis provided strong evidence (P_experiment= 0.0314) for a QTL affecting live weight. The QTL for live weight maps between markers BM1258 and BoLA DRBP1. A QTL significant at genome level (P_genome= 0.0087) was mapped for veterinary treatment, and the putative QTL probably affects susceptibility to milk fever or ketosis. In addition, three traits exceeded the chromosome 5% significance threshold: protein percentage of milk, calf mortality (sire), and milking speed. In within-family analyses, protein percentage was associated with markers in one family (LOD score = 4.5). Received: 14 December 1998 / Accepted: 28 March 1998     

Identification of quantitative trait loci associated with egg quality, egg production, and body weight in an F2 resource population of chickens

Egg production and egg quality are complex sex-limited traits that may benefit from the implementation of marker-assisted selection. The primary objective of the current study was to identify quantitative trait loci (QTL) associated with egg traits, egg production, and body weight in a chicken resource population. Layer (White Leghorn hens) and broiler (Cobb-Cobb roosters) lines were crossed to generate an F2 population of 508 hens over seven hatches. Phenotypes for 29 traits (weekly body weight from hatch to 6 weeks, egg traits including egg, albumen, yolk, and shell weight, shell thickness, shell puncture score, percentage of shell, and egg shell colour at 35 and 55 weeks of age, as well as egg production between 16 and 55 weeks of age) were measured in hens of the resource population. Genotypes of 120 microsatellite markers on 28 autosomal groups were determined, and interval mapping was conducted to identify putative QTL. Eleven QTL tests representing two regions on chromosomes 2 and 4 surpassed the 5% genome-wise significance threshold. These QTL influenced egg colour, egg and albumen weight, percent shell, body weight, and egg production. The chromosome 4 QTL region is consistent with multiple QTL studies that define chromosome 4 as a critical region significantly associated with a variety of traits across multiple resource populations. An additional 64 QTL tests surpassed the 5% chromosome-wise significance threshold.     

A general method for controlling the genome-wide type I error rate in linkage and association mapping experiments in plants

Control of the genome-wide type I error rate (GWER) is an important issue in association mapping and linkage mapping experiments. For the latter, different approaches, such as permutation procedures or Bonferroni correction, were proposed. The permutation test, however, cannot account for population structure present in most association mapping populations. This can lead to false positive associations. The Bonferroni correction is applicable, but usually on the conservative side, because correlation of tests cannot be exploited. Therefore, a new approach is proposed, which controls the genome-wide error rate, while accounting for population structure. This approach is based on a simulation procedure that is equally applicable in a linkage and an association-mapping context. Using the parameter settings of three real data sets, it is shown that the procedure provides control of the GWER and the generalized genome-wide type I error rate (GWER(k)).     

The number of loci that affect milk production traits in dairy cattle

We have used the results of an experiment mapping quantitative trait loci (QTL) affecting milk yield and composition to estimate the total number of QTL affecting these traits. We did this by estimating the number of segregating QTL within a half-sib daughter design using logic similar to that used to estimate the "false discovery rate" (FDR). In a half-sib daughter design with six sire families we estimate that the average sire was heterozygous for approximately 5 QTL per trait. Also, in most cases only one sire was heterozygous for any one QTL; therefore at least 30 QTL were likely to be segregating for these milk production traits in this Holstein population.     

Leveraging auxiliary data from arbitrary distributions to boost GWAS discovery with Flexible cFDR

Genome-wide association studies (GWAS) have identified thousands of genetic variants that are associated with complex traits. However, a stringent significance threshold is required to identify robust genetic associations. Leveraging relevant auxiliary covariates has the potential to boost statistical power to exceed the significance threshold. Particularly, abundant pleiotropy and the non-random distribution of SNPs across various functional categories suggests that leveraging GWAS test statistics from related traits and/or functional genomic data may boost GWAS discovery. While type 1 error rate control has become standard in GWAS, control of the false discovery rate can be a more powerful approach. The conditional false discovery rate (cFDR) extends the standard FDR framework by conditioning on auxiliary data to call significant associations, but current implementations are restricted to auxiliary data satisfying specific parametric distributions, typically GWAS p-values for related traits. We relax these distributional assumptions, enabling an extension of the cFDR framework that supports auxiliary covariates from arbitrary continuous distributions (“Flexible cFDR”). Our method can be applied iteratively, thereby supporting multi-dimensional covariate data. Through simulations we show that Flexible cFDR increases sensitivity whilst controlling FDR after one or several iterations. We further demonstrate its practical potential through application to an asthma GWAS, leveraging various functional genomic data to find additional genetic associations for asthma, which we validate in the larger, independent, UK Biobank data resource.     

A quick method for computing approximate thresholds for quantitative trait loci detection

This article proposes a quick method for computing approximate threshold levels that control the genome-wise type I error rate of tests for quantitative trait locus (QTL) detection in interval mapping (IM) and composite interval mapping (CIM). The procedure is completely general, allowing any population structure to be handled, e.g., BC(1), advanced backcross, F(2), and advanced intercross lines. Its main advantage is applicability in complex situations where no closed form approximate thresholds are available. Extensive simulations demonstrate that the method works well over a range of situations. Moreover, the method is computationally inexpensive and may thus be used as an alternative to permutation procedures. For given values of the likelihood-ratio (LR)-profile, computations involve just a few seconds on a Pentium PC. Computations are simple to perform, requiring only the values of the LR statistics (or LOD scores) of a QTL scan across the genome as input. For CIM, the window size and the position of cofactors are also needed. For the approximation to work well, it is suggested that scans be performed with a relatively small step size between 1 and 2 cM.