Skip to main content

Advertisement

SpringerLink
Log in

Swift block-updating EM and pseudo-EM procedures for Bayesian shrinkage analysis of quantitative trait loci

  • Original Paper
  • Published:
Theoretical and Applied Genetics Aims and scope Submit manuscript

Abstract

Introduction

Virtually all existing expectation-maximization (EM) algorithms for quantitative trait locus (QTL) mapping overlook the covariance structure of genetic effects, even though this information can help enhance the robustness of model-based inferences.

Results

Here, we propose fast EM and pseudo-EM-based procedures for Bayesian shrinkage analysis of QTLs, designed to accommodate the posterior covariance structure of genetic effects through a block-updating scheme. That is, updating all genetic effects simultaneously through many cycles of iterations.

Conclusion

Simulation results based on computer-generated and real-world marker data demonstrated the ability of our method to swiftly produce sensible results regarding the phenotype-to-genotype association. Our new method provides a robust and remarkably fast alternative to full Bayesian estimation in high-dimensional models where the computational burden associated with Markov chain Monte Carlo simulation is often unwieldy. The R code used to fit the model to the data is provided in the online supplementary material.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  • Ball RD (2001) Bayesian methods for quantitative trait loci mapping based on model selection: approximate analysis using Bayesian information criterion. Genetics 159:1351–1364

    PubMed  CAS  Google Scholar 

  • Bishop CM, Tipping ME (2003) Bayesian regression and classification. In: Suykens J, Horvath G, Basu S, Micchelli C, Vandewalle J (eds) Advances in learning theory: methods, models and applications, vol 190. IOS Press, NATO Science, Amsterdam, pp 267–285

    Google Scholar 

  • Broman KW, Speed TP (2002) A model selection approach for the identification of quantitative trait loci in experimental crosses (with discussion). J Roy Stat Soc B 64:641–656

    Article  Google Scholar 

  • Broman KW (2001) Review of statistical methods for QTL mapping in experimental crosses. Lab Anim 30:44–52

    CAS  Google Scholar 

  • Cai X, Huang A, Xu S (2011) Fast empirical Bayesian LASSO for multiple quantitative trait locus mapping. BMC Bioinform 12:211

    Article  Google Scholar 

  • Carbonell EA, Asins MJ, Baselga M, Balansard E, Gerig TM (1993) Power studies in the estimation of genetic parameters and the localization of quantitative trait loci for backcross and doubled haploid populations. Theor Appl Genet 86:411–416

    Article  Google Scholar 

  • Carlborg Ö, Andersson L (2002) Use of randomization testing to detect multiple epistatic QTLs. Genet Sel Evol 79:175–184

    Google Scholar 

  • Churchill GA, Doerge RW (1994) Empirical threshold values for quantitative trait mapping. Genetics 138:963–971

    PubMed  CAS  Google Scholar 

  • Cleveland MA, Forni S, Nader D, Maltecca C (2010) Genomic breeding value prediction using three Bayesian methods and application to reduced density marker panels. BMC Proc 4(Suppl 1):S6

    Article  PubMed  Google Scholar 

  • Conti DV, Witte J (2003) Hierarchical modeling of linkage disequilibrium: genetic structure and spatial relations. Am J Hum Genet 72:351–363

    Article  PubMed  CAS  Google Scholar 

  • de los Campos G, Naya H, Gianola D, Crossa J, Legarra A, Manfredi E, Weigel K, Cotes JM (2009) Predicting quantitative traits with regression models for dense molecular markers and pedigree. Genetics 182:375–385

    Article  Google Scholar 

  • Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm. J Roy Stat Soc B 39:1–38

    Google Scholar 

  • Fridley BL, Jenkins GD (2010) Localizing putative markers in genetic association studies by incorporating linkage disequilibrium into Bayesian hierarchical models. Hum Hered 70:63–73

    Article  PubMed  Google Scholar 

  • Gelman A, Hill J (2007) Data analysis using regression and multilevel/hierarchical models. Cambridge University Press, New York

    Google Scholar 

  • Gelman A, Carlin JB, Stern HS, Rubin DB (2003) Bayesian data analysis, 2nd edn. Chapman and Hall, New York

    Google Scholar 

  • George EI, McCulloch RE (1993) Variable selection via Gibbs sampling. J Am Stat Assoc 88:881–889

    Article  Google Scholar 

  • Gilks WR, Richardson S, Spiegelhalter DJ (eds) (1996) Markov Chain Monte Carlo in practice. Chapman and Hall, London

    Google Scholar 

  • Gimelfarb A, Lande R (1994a) Simulation of marker-assisted selection in hybrid populations. Genet Res 63:39–47

    Article  PubMed  CAS  Google Scholar 

  • Gimelfarb A, Lande R (1994b) Simulation of marker-assisted selection for non-additive traits. Genet Res 64:127–136

    Article  PubMed  CAS  Google Scholar 

  • Golub G, van Loan C (1996) Matrix computations, 3rd edn. The John Hopkins University Press, Baltimore

    Google Scholar 

  • Hayashi T, Iwata H (2010) EM algorithm for Bayesian estimation of genomic breeding values. BMC Genet 11:3

    Article  PubMed  Google Scholar 

  • Heckerman D, Chickering DM, Meek C, Rounthwaite R, Kadie C (2000) Dependency network for inference, collaborative filtering, and data visualization. J Mach Learn Res 1:49–75

    Google Scholar 

  • Henderson CR (1950) Estimation of genetic parameters. Ann Math Stat 21:309–310

    Google Scholar 

  • Henderson CR (1970) Best linear unbiased estimation and prediction under a selection model. Biometrics 31:423–447

    Article  Google Scholar 

  • Hoerl AE, Kennard RW (1970) Ridge regression: biased estimation for nonorthogonal problems. Technometrics 12:55–67

    Article  Google Scholar 

  • Hoti F, Sillanpää MJ (2006) Bayesian mapping of genotype × expression interactions in quantitative and qualitative traits. Heredity 97:4–18

    Article  PubMed  CAS  Google Scholar 

  • Jackson CH, Best NG, Richardson S (2009) Bayesian graphical models for regression on multiple data sets with different variables. Biostatistics 10:335–351

    Article  PubMed  CAS  Google Scholar 

  • Jeffreys H (1961) Theory of probability. Clarendon Press, Oxford

    Google Scholar 

  • Kabán A (2007) On Bayesian classification with Laplace priors. Patt Rec Lett 28:1271–1282

    Article  Google Scholar 

  • Kao C-H, Zeng Z-B, Teasdale RD (1999) Multiple interval mapping for quantitative trait loci. Genetics 152:1203–1216

    PubMed  CAS  Google Scholar 

  • Kass RE, Raftery AE (1995) Bayes factors. J Am Stat Assoc 90:773–795

    Article  Google Scholar 

  • Knürr T, Läärä E, Sillanpää MJ (2011) Genetic analysis of complex traits via Bayesian variable selection: the utility of a mixture of uniform priors. Genet Res 93:303–318

    Article  Google Scholar 

  • Kärkkäinen HP, Sillanpää MJ (2012) Back to basics for Bayesian model building in genomic selection. Genetics 191:969–987

    Google Scholar 

  • Lande R, Thompson R (1990) Efficiency of marker assisted selection in the improvement of quantitative traits. Genetics 124:743–756

    PubMed  CAS  Google Scholar 

  • Li Y, Campbell C, Tipping ME (2002) Bayesian automatic relevance determination algorithms for classifying gene expression data. Bioinformatics 18:1332–1339

    Article  PubMed  CAS  Google Scholar 

  • Li Z, Sillanpää MJ (2012a) Estimation of quantitative trait locus effects with epistasis by variational Bayes algorithms. Genetics 190:231–249

    Article  PubMed  Google Scholar 

  • Li Z, Sillanpää MJ (2012b) Overview of LASSO-related penalized regression methods for quantitative trait mapping and genomic selection. Theor Appl Genet 125:419–435

    Google Scholar 

  • Lowd D, Shamaei A (2011) Mean field inference in dependency networks: an empirical study. In: Proceedings of the 25th conference on artificial intelligence (AAAI-11), San Francisco, CA

  • Lunn D, Best N, Spiegelhalter D, Graham G, Neuenschwander B (2009) Combining MCMC with ‘sequential’ PKPD modelling. J Pharmacokinet Pharmacodyn 36:19–38

    Article  PubMed  CAS  Google Scholar 

  • Makhuvha T, Pegram G, Sparks R, Zucchini W (1997) Patching rainfall data using regression methods. 1. Best subset selection, EM and pseudo-EM methods: theory. J Hydrol 198:289–307

    Article  Google Scholar 

  • Malo N, Libiger O, Schork NJ (2008) Accommodating linkage disequilibrium in genetic-association analyses via ridge regression. Am J Hum Genet 82:375–385

    Article  PubMed  CAS  Google Scholar 

  • McLachlan GJ, Krishnan T (1997) The EM algorithm and extensions. Wiley, New York

    Google Scholar 

  • Meuwissen THE, Hayes BJ, Goddard ME (2001) Prediction of total genetic value using genome-wide dense marker maps. Genetics 157:1819–1829

    PubMed  CAS  Google Scholar 

  • Mutshinda CM, O’Hara RB, Woiwod IP (2011) A multispecies perspective on ecological impacts of climatic forcing. J Anim Ecol 80:101–107

    Article  PubMed  Google Scholar 

  • Mutshinda CM, Sillanpää MJ (2011) Bayesian shrinkage analysis of QTLs under shape-adaptive shrinkage priors, and accurate re-estimation of genetic effects. Heredity 107:405–412

    Article  PubMed  CAS  Google Scholar 

  • Mutshinda CM, Sillanpää MJ (2010) Extended Bayesian LASSO for multiple quantitative trait loci mapping and unobserved phenotype prediction. Genetics 186:1067–1075

    Article  PubMed  Google Scholar 

  • Mutshinda CM, O’Hara RB, Woiwod IP (2009) What drives community dynamics? Proc R Soc B 276:2923–2929

    Article  PubMed  Google Scholar 

  • Miller A (2002) Subset selection in regression. Chapman and Hall, London

    Book  Google Scholar 

  • Myers RL (1992) Classical and modern regression analysis, 2nd edn. Wiley, New-York

    Google Scholar 

  • O’Hara RB, Sillanpää MJ (2009) A review of Bayesian variable selection methods: what, how and which. Bayesian Anal 4:85–118

    Article  Google Scholar 

  • R Development Core Team (2011) R: A language and environment for statistical computing, reference index version 2.13.2. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0. http://www.R-project.org

  • Sen S, Churchill GA (2001) A statistical framework for quantitative trait mapping. Genetics 159:371–387

    PubMed  CAS  Google Scholar 

  • Shepherd R, Meuwissen THE, Woolliams JA (2010) Genomic selection and complex trait prediction using a fast EM algorithm applied to genome-wide markers. BMC Bioinform 11:529

    Article  Google Scholar 

  • Sillanpää MJ, Hoti F (2007) Mapping quantitative trait loci from a single tail sample of the phenotype distribution including survival data. Genetics 177:2361–2377

    Article  PubMed  Google Scholar 

  • Sillanpää MJ, Bhattacharjee M (2006) Association mapping of complex trait loci with context-dependent effects and unknown context-variable. Genetics 174:1597–1611

    Article  PubMed  Google Scholar 

  • Sillanpää MJ, Bhattacharjee M (2005) Bayesian association-based fine mapping in small chromosomal segments. Genetics 169:427–439

    Article  PubMed  Google Scholar 

  • Sillanpää MJ, Corander J (2002) Model choice in gene mapping: what and why. Trends Genet 18:301–307

    Article  PubMed  Google Scholar 

  • Stephens M, Smith NJ, Donnelly P (2001) A new statistical method for haplotype reconstruction from population data. Am J Hum Genet 68:978–989

    Article  PubMed  CAS  Google Scholar 

  • Sun W, Ibrahim JG, Zou F (2010) Genome-wide multiple loci mapping in experimental crosses by the iterative penalized regression. Genetics 185:349–359

    Article  PubMed  CAS  Google Scholar 

  • ter Braak CJF, Boer MP, Bink MCAM (2005) Extending Xu’s Bayesian model for estimating polygenic effects using markers of the entire genome. Genetics 170:1435–1438

    Article  PubMed  Google Scholar 

  • Tibshirani R (1996) Regression shrinkage and selection via LASSO. J Roy Stat Soc B 58:267–288

    Google Scholar 

  • Tinker NA, Mather DE, Rosnagel BG, Kasha KJ, Kleinhofs A (1996) Regions of the genome that affect agronomic performance in two-row barley. Crop Sci 36:1053–1062

    Article  Google Scholar 

  • Tipping ME, Lawrence ND (2005) Variational inference for Student-t models: robust Bayesian interpolation and generalized component analysis. NeuroComputing 69:123–141

    Article  Google Scholar 

  • Tipping ME (2001) Sparse Bayesian learning and the relevance vector machine. J Mach Learn Res 1:211–244

    Google Scholar 

  • Wang S, Basten CJ, Zeng Z-B (2006) Windows QTL Cartographer 2.5. Department of Statistics, North Carolina State University, Raleigh, NC

  • Wang H, Zhang Y-M, Li X, Masinde GL, Mohan S, Baylink DJ, Xu S (2005) Bayesian shrinkage estimation of quantitative trait loci parameters. Genetics 170:465–480

    Article  PubMed  CAS  Google Scholar 

  • Whittaker JC, Thompson R, Denham MC (2000) Marker-assisted selection using ridge regression. Genet Res 75:249–252

    Article  PubMed  CAS  Google Scholar 

  • Xu S (2010) An expectation-maximization algorithm for the Lasso estimation of quantitative trait locus effects. Heredity 105:483–494

    Article  PubMed  CAS  Google Scholar 

  • Xu S (2007) An empirical Bayes method for estimating epistatic effects of quantitative trait loci. Biometrics 63:513–521

    Article  PubMed  CAS  Google Scholar 

  • Xu S (2003) Estimating polygenic effects using markers of the entire genome. Genetics 163:789–801

    PubMed  CAS  Google Scholar 

  • Xu S, Jia Z (2007) Genomewide analysis of epistatic effects for quantitative traits in barley. Genetics 175:1955–1963

    Article  PubMed  CAS  Google Scholar 

  • Yi N, Banerjee S (2009) Hierarchical generalized linear models for multiple quantitative trait locus mapping. Genetics 181:1101–1113

    Article  PubMed  CAS  Google Scholar 

  • Yi N, Xu S (2008) Bayesian LASSO for quantitative trait loci mapping. Genetics 179:1045–1055

    Article  PubMed  CAS  Google Scholar 

  • Yi N, George V, Allison DB (2003) Stochastic search variable selection for identifying multiple quantitative trait loci. Genetics 164:1129–1138

    PubMed  CAS  Google Scholar 

  • Yi N, Shriner D, Banerjee S, Mehta T, Pomp D, Yandell BS (2007) An efficient Bayes model selection approach for interacting quantitative trait loci models with many effects. Genetics 176:1865–1877

    Article  PubMed  Google Scholar 

  • Zielke G (1968) Inversion of modified symmetric matrices. J Assoc Comput Mach 15:402–408

    Article  Google Scholar 

  • Zou H, Hastie T (2005) Regularization and variable selection via the elastic net. J Roy Stat Soc B 67:301–320

    Article  Google Scholar 

Download references

Acknowledgments

The authors wish to thank Hanni Kärkkäinen, Zitong Li and two anonymous referees for their pertinent comments and suggestions. This work was supported by a research grants from the Academy of Finland and University of Helsinki’s research funds.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Helsinki, PO Box 68, 00014, Helsinki, Finland

    Crispin M. Mutshinda & Mikko J. Sillanpää

  2. Department of Agricultural Sciences, University of Helsinki, PO Box 27, 00014, Helsinki, Finland

    Mikko J. Sillanpää

  3. Department of Mathematical Sciences, University of Oulu, PO Box 3000, 90014, Oulu, Finland

    Mikko J. Sillanpää

  4. Department of Biology, University of Oulu, PO Box 3000, 90014, Oulu, Finland

    Mikko J. Sillanpää

  5. Department of Mathematics and Computer Science, Mount Allison University, 67 York Street, Sackville, New Brunswick, E4L 1E6, Canada

    Crispin M. Mutshinda

Authors
  1. Crispin M. Mutshinda
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mikko J. Sillanpää
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mikko J. Sillanpää.

Additional information

Communicated by F. van Eeuwijk.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (PDF 177 kb)

Supplementary material 2 (PDF 17 kb)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mutshinda, C.M., Sillanpää, M.J. Swift block-updating EM and pseudo-EM procedures for Bayesian shrinkage analysis of quantitative trait loci. Theor Appl Genet 125, 1575–1587 (2012). https://doi.org/10.1007/s00122-012-1936-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00122-012-1936-1

Keywords

Search

Navigation