- Methodology Article
- Open Access
Advantages of continuous genotype values over genotype classes for GWAS in higher polyploids: a comparative study in hexaploid chrysanthemum
- Fabian Grandke^{1, 2}Email authorView ORCID ID profile,
- Priyanka Singh^{1, 3},
- Henri C. M. Heuven^{1, 4},
- Jorn R. de Haan^{1} and
- Dirk Metzler^{2}
- Received: 19 February 2016
- Accepted: 11 July 2016
- Published: 24 August 2016
Abstract
Background
Association studies are an essential part of modern plant breeding, but are limited for polyploid crops. The increased number of possible genotype classes complicates the differentiation between them. Available methods are limited with respect to the ploidy level or data producing technologies. While genotype classification is an established noise reduction step in diploids, it gains complexity with increasing ploidy levels. Eventually, the errors produced by misclassifications exceed the benefits of genotype classes. Alternatively, continuous genotype values can be used for association analysis in higher polyploids. We associated continuous genotypes to three different traits and compared the results to the output of the genotype caller SuperMASSA. Linear, Bayesian and partial least squares regression were applied, to determine if the use of continuous genotypes is limited to a specific method. A disease, a flowering and a growth trait with h ^{2} of 0.51, 0.78 and 0.91 were associated with a hexaploid chrysanthemum genotypes. The data set consisted of 55,825 probes and 228 samples.
Results
We were able to detect associating probes using continuous genotypes for multiple traits, using different regression methods. The identified probe sets were overlapping, but not identical between the methods. Baysian regression was the most restrictive method, resulting in ten probes for one trait and none for the others. Linear and partial least squares regression led to numerous associating probes. Association based on genotype classes resulted in similar values, but missed several significant probes. A simulation study was used to successfully validate the number of associating markers.
Conclusions
Association of various phenotypic traits with continuous genotypes is successful with both uni- and multivariate regression methods. Genotype calling does not improve the association and shows no advantages in this study. Instead, use of continuous genotypes simplifies the analysis, saves computational time and results more potential markers.
Keywords
- Association study
- Polyploids
- Linear regression
- Bayz
- Partial least squares
- Continuous genotypes
Background
Many agriculturally and horticulturally important crops are polyploid [1, 2]. Polyploids have multiple sets of chromosomes and have arisen by extensive genomic alteration and genome duplication [3, 4]. Diploidization, the differentiation of duplicated loci, converts most polyploids back to diploids on the long term [5, 6]. The phenomenon of polyploidy results in complex genomic architecture in many flowering plants thus complicates genomics-based breeding [7]. Research in polyploids is also limited by the available methods and technologies [8]. Most bioinformatic tools have been developed for diploids and cannot be applied to higher ploidy levels. Recently, several methods have been developed to overcome this limitation, but most of them are restricted to tetraploids [9, 10]. Association studies aim to determine a genetic origin for a phenotypic trait [11]. While phenotyping is independent of the ploidy level, genotyping has been identified as a bottleneck in breeding of polyploid crops [12]. Genotyping describes the process of determining an organism’s genotype [13] and is known to be erroneous [14]. It involves the extraction of genetic material, molecular biological processes and the assignment of genotypic classes.
Generally, determination of genotype classes (genotype calling) reduces noise and leads to better associations. In polyploids, this task is not as straightforward and the advantage of noise reduction is reversed by the high risk of misclassification, i.e. assignment of wrong genotypes to samples. Therefore, we skipped genotype calling and used the continuous genotypic values (compare Eq. 6) directly. The aim of this study was to use these continuous values to detect probes associating to three traits in hexaploid chrysanthemum and compare the results to genotype classes to evaluate the advantage of our approach. The traits have been selected to represent distinctive types (disease resistance, flowering and growth) and heritabilities (0.51, 0.78 and 0.92). We applied linear regression (LR), bayz (Bayesian regression) [21, 22] partial least squares regression (PLSR) [23, 24] and compared the results to avoid methodological bias. We showed that the assignment of genotype classes would not improve our findings, but lead to misclassification. In this article we demonstrate that we are able to use continuous genotype values to detect associations with three different traits in hexaploid chrysanthemum.
Results and discussion
Disease trait
Additional file 2 shows a comparison of the significance values by SuperMASSA and the continuous values. Taken together, genotype calling distorted genotypic INFORMATION for some markers and the prevented their correctassociation. The desired noise reduction, which improves associations, could not be achieved. PLSR detected 83 probes with variance importance (VIM) scores ≥2. Three of them overlap with the LR results. bayz did not find any significant probes with a Bayes factor (BF) threshold over 10. Even with a less-strict threshold of 5 there were no findings (compare Additional file 3).
Flowering trait
LR detected 439 significant probes for the flowering trait with the continuous values. SuperMASSA genotype calls led to 332 significant probes (Additional file 4). Again, the continuous genotypes generally lead to lower q-values than the genotype calls (compare Additional file 2). We simulated the experiment with varying numbers of significant probes (2-10) 100 times each. That way, we could observe how many significant probes we would detect if the true genotypes are known. The simulation results are shown in Additional file 5 and show that we expect around 100-2000 significant probes. Hence, our association results are in the correct magnitude. For our dataset continuous genotype values are advantageous over genotype classes because we obtain more significant probes and are less likely to miss trait related probes. However, our method is not too insensitive and does not result in thousands of false positive markers. In fact, the simulation study detected even more false positives. It reduces the number of potential candidate probes from 55,825 to 439.
An example probe for the flowering trait is shown in Fig. 3 d-f. SuperMASSA identified three different genotype classes. Continuous genotype values indicate four genotype classes, roughly centered at 0.75, 1.5, 2.2 and 3.5 (Fig. 3 e). In contrast, the genotype classes by SuperMASSA combine the first two clusters (Fig. 3 d). The blue one contains samples of the first and third cluster, which leads to its spread over the whole DEBV range (Fig. 3 f). This leads to a lower p-value, but the probe is still highly significant. Nevertheless, in other cases this difference might determine whether the null hypothesis can be rejected or not.
The probes were selected so that not more than three SNPs (six probes) lay on one contig. Thus, it is unlikely to detect two or more significant SNPs from the same contig by chance. The fact that we found up to three SNPs from the same contig is a good indicator for real association. We detected 5 probes from one contig, coding for three different SNPs. The white area in the second bar in Fig. 4 shows 28 single probes, which code for different markers, but are located on the same contig. Accordingly, the missing second probe is not required to proof association.
Overlapping significant markers between all three methods for the flowering trait
Marker | R ^{2} | q-value | BF | VIM |
---|---|---|---|---|
AX-89300609 | 0.31 | 1.72×10^{−15} | 11.03 | 3.23 |
AX-89215144 | 0.26 | 8.90×10^{−13} | 15.49 | 2.90 |
AX-89213862 | 0.18 | 1.74×10^{−8} | 24.82 | 4.12 |
AX-89256548 | 0.09 | 5.16×10^{−6} | 12.27 | 3.45 |
Growth trait
LR did not result in any significant probes for the growth trait. Growth is known to be a polygenic trait [25]. Hence, we did not expect single probes to show strong association. The association with the SuperMASSA genotype calls did not output any significant probes, either (Additional file 6). An example probe is shown in Fig. 3 g. All but five samples were assigned the same genotype class. We expect monomorphic probes, but in Fig. 3 h we see that the genotype values span a large portion of the negative θ range. Thus, we expect multiple genotype classes within that cloud of data points. However, they could neither be determined manually nor computationally. bayz did not detect any significant probes, either. In contrast, the PLSR association detected 62 significant probes. It includes low effect markers as well and is not limited to single loci. For polygenic traits PLSR is therefore advantageous.
General
We could have applied sparse partial least squares (SPLS) regression to deal with very high-dimensional data. The expected number of associating probes would be even higher than. We decided against it because the number of false positives and low impact probes (based on explained variance from the LR analysis) would have increased significantly. PLSR is less restrictive in the growing and disease trait but not in the flowering trait.
EBVs are established metrics in breeding, but have been criticized for genome-wide association studies (GWAS) because their naïve usage reduces power, increases the false positive rate and misestimates effect sized of quantitative trait loci (QTL) [26]. Hence, we used deregressed EBVs to account for fixed effects and repeated measurements, as described by Garrick et al. [27].
SuperMASSA misclassifies genotypes in some cases because it assumes clusters of equal distance at fixed positions. Instead, the data does not necessarily segregate into clusters (Fig. 3 g). If clusters can be identified, they are not always at fixed positions, as assumed by SuperMASSA (Fig. 3 d). Further, it does not account for outliers and assigns genotype classes to every sample. This results in clusters of only one sample in some cases and does not represent the genotype. Taken together, SuperMASSA does not cluster the data points properly for all probes. It seems to be optimized for data produced with two technologies and therefore performs less well on other data sets. The same situation was described by Voorips et al. [16], when they compared fitTetra and beadarrayMSV. Thus, genotype calling has no advantage for polyploid data generated with the Affymetrix Axiom™ technology, as long as no method works properly. This underpins our preliminary analyis, which showed that the resolution of the signal intensities is not large enough to distiguish between the increased number of genotype classes expected in hexaploids (Fig. 1). To this end, association of the continuous genotypes is currently the best method.
where G is the set of alleles and G _{ i } is the read count of allele i. Alternatives to the log-transformations might be more effective and need to be investigated [17].
The higher the ploidy level, the smaller is the advantage of genotype classes over the continuous values. With increasing numbers of genotype classes, the effect of noise reduction declines. For instance, in a diploid we expect the clusters AA, AB and BB at around 2, 0 and −2, respectively. A value of 1.2 would be assigned to cluster AA, so we correct for a large proportion of the signal. For any ploidy level n we expect up to 2n+1 clusters on a similar range, because the overall signal strength is limited by the used technology (e.g. amount of genetic material, GBS read depth). Consequently the distances between clusters decrease and the correction accounts for smaller proportions of the signal. In addition, the risk of misclassifications increases, because there is less tolerance for variation in the signal intensity or clusters overlap. Further, the distribution of genotype values approximates a continuous distribution with increasing ploidy levels. Figure 1 shows an example of a simulated hexaploid marker and one from the real data set. We were not able to identify the genotype classes for all samples in that case, because the clusters are indistinguishable. Nevertheless, the data points spread over the whole range of θ and provide genotypic information. From a biological perspective, continuous genotypes are difficult to interpret, because the number of alleles is discrete and should fall in one of the genotype classes. One explanation are tri- and tetra-allelic SNPs, where more than two nucleotides are present at the same position [29, 30]. If they are measured with bi-allelic technology (e.g. genotyping arrays), the sum of the two allele counts does not necessarily add up to the expected number (ploidy level). Alternatively, we might observe fractionation, the deletions in sub-genomes of allopolyploids [31]. Both result in data points outside of the expected clusters. For the association we mean-centered the genotype values.
Skipping genotype calling leads to further challenges with current linkage mapping and haplotype phasing methods, because they require genotype classes. Nevertheless, the choice of tools that work for polyploids is very limited anyways and new solutions need to be developed. Further, low-coverage sequencing and imputations of genotypes add more difficulties [32].
Conclusions
We showed that continuous genotype data can be used successfully in an association study of a polyploid crop and validated our findings in a simulation study. Application of different regression tools show that our approach is not limited to a specific method, but the results vary to a large extend.
Genotype calling leads to misclassification and false association results in some cases, where significant markers could not be detected. However, the majority of markers lead to similar results with genotype classes and continuous values, indicating that genotype calling is not adversely in general. In this study genotype calling has no advantage and can be skipped unless better methods are developed. Instead, use of continuous genotypes simplifies the analysis, saves computational time and results more potential markers. Nevertheless, the overlapping clusters of the given data set remain a challenge and the use of continuous genotypes is a successful solution to that problem.
Methods
A hexaploid chrysanthemum population consisting of 228 F1 offspring was used for our study. The cultivated plant material was provided by Dümmen Orange and all experiments have been performed according to legal guidelines.
Phenotypes
Overview of traits (DEBV)
Trait | h ^{2} | Mean | SD | Range | Samples |
---|---|---|---|---|---|
Disease | 0.51 | 0.05 | 1.2 | 6.19 | 228 |
Flowering | 0.78 | 0.00 | 3.3 | 19.01 | 228 |
Growth | 0.92 | 0.21 | 12.2 | 63.57 | 228 |
where \({r_{i}^{2}}\) is the reliability of the EBV of plant i and \({\sigma _{g}^{2}}\) is the additive genetic variance.
Genotypes
An example of a bi-allelic probe from a hexaploid chrysanthemum data set is shown in Fig. 1. The x-axis represents θ, the difference between the two alleles A and B. The values span the whole range of potential genotypes and represent seven different genotype classes. The leftmost samples are homozygous A, while the rightmost ones are homozygous B. The intermediates are heterozygous in varying proportions. The y-axis shows the mean signal strength s. The homozygous s values are lower, because logarithmic values are used.
The genotype calling was done with the web application of SuperMASSA without population-level information (http://statgen.esalq.usp.br/SuperMASSA/) [12]. The ploidy range was set from 2 to 6; the other parameters were used with the default parameters. We used the raw values of the two alleles as input. The resulting genotypes represented the numbers of the two alleles. For the association we used the difference between the counts of the first and second allele.
Association methods
where Y _{ i } are the DEBV s, α is the population mean, β is the regression coefficient, x _{ i } the mean-centered, continuous genotype value and ε _{ i } the residual error. The function lm from the R package stats (Version 3.1.3) with the default parameters was used for the regression [35]. The resulting p-values were transformed into q-values with the function qvalue of the R-package qvalue (Version 1.43.0) with default parameters [36, 37]. We applied a threshold of 0.01 to select the significantly associating probes.
Where the ‘null’ distribution modeled the majority of SNP with (virtually) no effect using prior settings π _{0}=0.99 and \(\sigma _{g0}^{2}=0.001\). The second distribution modeled SNPs with large effects where prior settings were π _{1}=0.01 and \(\sigma ^{2}_{g1}=0.1\). Variances of the mixture distribution and other model effects were estimated using a uniform prior and sampled with a Monte Carlo Markov Chain (MCMC) using Gibbs sampling. The MCMC was run for 50,000 iterations with a burn-in of 10,000 iterations and a thin-interval of 200. A Bernoulli distribution specified probabilities for a SNP belonging to the ‘null’ or second distribution and proportions for the mixture were set to have a slightly informative prior distribution ∼B e t a(10,1).
Where n is the number of samples, γ _{ i } is the observed and \(\hat {\gamma }_{i}\) is the predicted phenotypic value. In the third step the model was build and the significant probes were predicted based on their variable importance measurement (VIM) scores with a lower threshold of 2 [41, 42]. The scores were determined with the varImp function from the R-package caret [40]. The calculation is based on the weighted sums of the absolute regression coefficients.
Simulation
where π _{ j } is the explained variance \(\left (\sum _{j}\pi _{j} \text { is the}\text {heritability}\right)\), f _{ j } is the allele frequency and a _{ ij } is the genotype of sample i for probe j. The heritability was set to 0.78 as for the flowering trait. Afterwards, we associated the simulated phenotypes with the genotypes using LR. This simulation procedure was repeated 100 times for each parameter combination.
Abbreviations
BF, Bayes factor; PLSR, partial least squares regression; DEBV, deregressed estimated breeding value; GBS, genotyping by sequencing; LR, linear regression; MCMC, Monte Carlo Markov Chain; PEV, predictive error variance; REML, residual maximum likelihood; SNP, single nucleotide polymorphism; SPLS, sparse partial least squares
Declarations
Acknowledgements
This project (INTERCROSSING) has received funding from the European Union’s Seventh Framework Programme for research, technological development and demonstration under grant agreement no 289974. The APC fee for this article has been funded by the EC FP7 Post-Grant Open Access Pilot. We thank Dümmen Orange for providing the data set and Pascal Duenk for assistance with the DEBV s and bayz. We acknowledge Luc Janssen for the suggestion of an appropriate model for handling continuous genotypes and the critical comments by two anonymous reviewers on an earlier version of the manuscript.
Availability of data and materials
The genotype data set supporting the results of this article is available in the Zenodo repository, http://zenodo.org/record/46285.
Authors’ contributions
FG did the data preprocessing, the LR analysis, simulation study and drafted the manuscript. HH adviced on the study design and helped to draft the manuscript. PS calculated the DEBV s and did the association with PLSR and bayz. JRH participated in the study design and coordination. DM reviewed the manuscript and designed the simulation study. All authors read and approved the final manuscript.
Authors’ information
Some authors are employed at Genetwister Technologies B.V. and the data is owned by Dümmen Orange, a shareholder of the company.
Competing interests
The authors declare that they have no competing interests.
Consent for publication
Not applicable.
Ethics approval and consent to participate
Not applicable.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
References
- Soltis DE, Soltis PS, Tate JA. Advances in the study of polyploidy since plant speciation. New Phytologist. 2003; 161(1):173–91.View ArticleGoogle Scholar
- Leitch AR, Leitch IJ. Genomic plasticity and the diversity of polyploid plants. Science. 2008; 320(5875):481–3.View ArticlePubMedGoogle Scholar
- Acquaah G. Principles of plant genetics and breeding. Malden: Wiley-Blackwell; 2012.View ArticleGoogle Scholar
- Comai L. The advantages and disadvantages of being polyploid. Nat Rev Genet. 2005; 6(11):836–46.View ArticlePubMedGoogle Scholar
- Wang X, Shi X, Hao B, Ge S, Luo J. Duplication and DNA segmental loss in the rice genome: implications for diploidization. New Phytologist. 2005; 165(3):937–46.View ArticlePubMedGoogle Scholar
- Paterson AH, Bowers JE, Chapman BA. Ancient polyploidization predating divergence of the cereals, and its consequences for comparative genomics. Proc Natl Acad Sci U S A. 2004; 101(26):9903–8.View ArticlePubMedPubMed CentralGoogle Scholar
- Ramsey J, Schemske DW. Pathways, mechanisms, and rates of polyploid formation in flowering plants. Annu Rev Ecol Syst. 1998; 29(1):467–501.View ArticleGoogle Scholar
- Soltis DE, Buggs RJA, Doyle JJ, Soltis PS. What we still don’t know about polyploidy. Taxon. 2010; 59(5):1387–403.Google Scholar
- Dufresne F, Stift M, Vergilino R, Mable BK. Recent progress and challenges in population genetics of polyploid organisms: an overview of current state-of-the-art molecular and statistical tools. Mol Ecol. 2014; 23(1):40–69.View ArticlePubMedGoogle Scholar
- Grandke F, Ranganathan S, Czech A, de Haan JR, Metzler D. Bioinformatic tools for polyploid crops. J Agric Sci Technol B. 2014; 4:593–601.Google Scholar
- Hirschhorn JN, Daly MJ. Genome-wide association studies for common diseases and complex traits. Nat Rev Genet. 2005; 6(2):95–108.View ArticlePubMedGoogle Scholar
- Serang O, Mollinari M, Garcia AAF. Efficient exact maximum a posteriori computation for bayesian SNP genotyping in polyploids. PLoS ONE. 2012; 7(2):30906.View ArticleGoogle Scholar
- Syvänen AC. Accessing genetic variation: genotyping single nucleotide polymorphisms. Nat Rev Genet. 2001; 2(12):930–42.View ArticlePubMedGoogle Scholar
- Pompanon F, Bonin A, Bellemain E, Taberlet P. Genotyping errors: causes, consequences and solutions. Nat Rev Genet. 2005; 6(11):847–6.View ArticlePubMedGoogle Scholar
- Lamy P, Grove J, Wiuf C. A review of software for microarray genotyping. Human Genomics. 2011; 5(4):304–9. 21712191.PubMedPubMed CentralGoogle Scholar
- Voorrips RE, Gort G, Vosman B. Genotype calling in tetraploid species from bi-allelic marker data using mixture models. BMC Bioinformatics. 2011; 12(1):172.View ArticlePubMedPubMed CentralGoogle Scholar
- Gidskehaug L, Kent M, Hayes BJ, Lien S. Genotype calling and mapping of multisite variants using an atlantic salmon iSelect SNP array. Bioinformatics. 2011; 27(3):303–10.View ArticlePubMedGoogle Scholar
- Ankerst M, Breunig MM, Kriegel HP, Sander J. OPTICS: Ordering points to identify the clustering structure. In: Proceedings of the 1999 ACM SIGMOD International Conference on Management of Data. Philadelphia: ACM Press: 1999. p. 49–60.Google Scholar
- Ester M, Kriegel H-P, Sander J, Xu X. A density-based algorithm for discovering clusters in large spatial databases with noise. In: Proceedings of the 2nd International Conference on Knowledge Discovery and Data Mining. Portland: AAAI Press: 1996. p. 226–31.Google Scholar
- Wang S, Wong D, Forrest K, Allen A, Chao S, Huang BE, Maccaferri M, Salvi S, Milner SG, Cattivelli L, Mastrangelo AM, Whan A, Stephen S, Barker G, Wieseke R, Plieske J, International Wheat Genome Sequencing Consortium, Lillemo M, Mather D, Appels R, Dolferus R, Brown-Guedira G, Korol A, Akhunova AR, Feuillet C, Salse J, Morgante M, Pozniak C, Luo MC, Dvorak J, Morell M, Dubcovsky J, Ganal M, Tuberosa R, Lawley C, Mikoulitch I, Cavanagh C, Edwards KJ, Hayden M, Akhunov E. Characterization of polyploid wheat genomic diversity using a high-density 90 000 single nucleotide polymorphism array. Plant Biotechnol J. 2014; 12(6):787–96.View ArticlePubMedPubMed CentralGoogle Scholar
- Heuven HCM, Janss LLG. Bayesian multi-QTL mapping for growth curve parameters. BMC Proc. 2010; 4:12. 20380755.View ArticleGoogle Scholar
- Kapell DN, Sorensen D, Su G, Janss LL, Ashworth CJ, Roehe R. Efficiency of genomic selection using Bayesian multi-marker models for traits selected to reflect a wide range of heritabilities and frequencies of detected quantitative traits loci in mice. BMC Genet. 2012; 13(1):42.View ArticlePubMedPubMed CentralGoogle Scholar
- Jöreskog KG, Wold HOA. Systems under indirect observation: causality, structure, prediction. Amsterdam: North-Holland; 1982.Google Scholar
- Kuhn M, Johnson K. Linear regression and its cousins. In: Applied Predictive Modeling. New York: Springer: 2013. p. 112–21.View ArticleGoogle Scholar
- Salas Fernandez MG, Becraft PW, Yin Y, Lübberstedt T. From dwarves to giants? Plant height manipulation for biomass yield. Trends Plant Sci. 2009; 14(8):454–61. doi:10.1016/j.tplants.2009.06.005.View ArticlePubMedGoogle Scholar
- Ekine CC, Rowe SJ, Bishop SC, de Koning D-J. Why breeding values estimated using familial data should not be used for genome-wide association studies. G3: Genes|Genomes|Genetics. 2013; 4(2):341–7. 24362310.View ArticlePubMed CentralGoogle Scholar
- Garrick DJ, Taylor JF, Fernando RL. Deregressing estimated breeding values and weighting information for genomic regression analyses. Genet Sel Evol. 2009; 41(1):55. 20043827.View ArticlePubMedPubMed CentralGoogle Scholar
- Hill WG, Goddard ME, Visscher PM. Data and theory point to mainly additive genetic variance for complex traits. PLoS Genet. 2008; 4(2):1000008.View ArticleGoogle Scholar
- Casci T. Population genetics: SNPs that come in threes. Nat Rev Genet. 2010; 11(1):8–8.View ArticlePubMedGoogle Scholar
- Phillips C, Amigo J, Carracedo A, Lareu MV. Tetra-allelic SNPs: informative forensic markers compiled from public whole-genome sequence data. Forensic Sci Int Genet. 2015; 19:100–6. 26209763.View ArticlePubMedGoogle Scholar
- Langham RJ, Walsh J, Dunn M, Ko C, Goff SA, Freeling M. Genomic duplication, fractionation and the origin of regulatory novelty. Genetics. 2004; 166(2):935–45. 15020478.View ArticlePubMedPubMed CentralGoogle Scholar
- Pasaniuc B, Rohland N, McLaren PJ, Garimella K, Zaitlen N, Li H, Gupta N, Neale BM, Daly MJ, Sklar P, Sullivan PF, Bergen S, Moran JL, Hultman CM, Lichtenstein P, Magnusson P, Purcell SM, Haas DW, Liang L, Sunyaev S, Patterson N, de Bakker PIW, Reich D, Price AL. Extremely low-coverage sequencing and imputation increases power for genome-wide association studies. Nat Genet. 2012; 44(6):631–5.View ArticlePubMedPubMed CentralGoogle Scholar
- Gilmour AR, Thompson R, Cullis BR. Average information REML: an efficient algorithm for variance parameter estimation in linear mixed models. Biometrics. 1995; 51(4):1440–50. doi:10.2307/2533274.View ArticleGoogle Scholar
- Affymetrix Power Tools. 2015. http://www.affymetrix.com/estore/partners_programs/\programs/developer/tools/powertools.affx. Accessed 25 Jul 2015.
- R Core Team. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing; 2015. http://www.r-project.org/. Accessed 13 Aug 2015.
- Storey JD. Qvalue: Q-value Estimation for False Discovery Rate Control. 2015. R package version 2.0.0. http://qvalue.princeton.edu/. Accessed 13 Aug 2015.
- Storey JD. The positive false discovery rate: a bayesian interpretation and the q-value. Ann Stat. 2003; 31(6):2013–35. 3448445.View ArticleGoogle Scholar
- Schurink A, Janss LL, Heuven HC. Bayesian Variable Selection to identify QTL affecting a simulated quantitative trait. BMC Proc. 2012; 6(Suppl 2):8. doi:10.1186/1753-6561-6-S2-S8.View ArticleGoogle Scholar
- Habier D, Fernando RL, Kizilkaya K, Garrick DJ. Extension of the bayesian alphabet for genomic selection. BMC Bioinformatics. 2011; 12(1):186.View ArticlePubMedPubMed CentralGoogle Scholar
- Kuhn M, Wing J, Weston S, Williams A, Keefer C, Engelhardt A. Caret: Classification and Regression Training. 2012. R package version 5.15-044. http://cran.r-project.org/package=caret.
- Mehmood T, Liland KH, Snipen L, Sæbø S. A review of variable selection methods in partial least squares regression. Chemometr Intell Lab Syst. 2012; 118:62–9.View ArticleGoogle Scholar
- Stephen Milborrow: Notes on the earth package. 2015. http://www.milbo.org/doc/earth-notes.pdf. Accessed 13 Aug 2015.
- Voorrips RE, Maliepaard CA. The simulation of meiosis in diploid and tetraploid organisms using various genetic models. BMC Bioinformatics. 2012; 13(1):248. Accessed 30 Apr 2014.View ArticlePubMedPubMed CentralGoogle Scholar
- Günther T, Gawenda I, Schmid KJ. phenosim - A software to simulate phenotypes for testing in genome-wide association studies. BMC Bioinformatics. 2011; 12(1):265. doi:10.1186/1471-2105-12-265. Accessed 2015-12-18.View ArticlePubMedPubMed CentralGoogle Scholar