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Vol. 59, No. 4, 2005
Issue release date: August 2005
Section title: Original Paper
Hum Hered 2005;59:210–219
(DOI:10.1159/000086699)

The Posterior Probability of Linkage Allowing for Linkage Disequilibrium and a New Estimate of Disequilibrium between a Trait and a Marker

Yang X.a, b · Huang J.a, c, e · Logue M.W.a, e · Vieland V.J.a, d, e
aCenter for Statistical Genetics Research, Departments of bBiostatistics, cStatistics and Actuarial Science, dPsychiatry, and ePublic Health Genetics, The University of Iowa, Iowa City, Iowa, USA
email Corresponding Author

Abstract

The posterior probability of linkage (PPL) statistic has been developed as a method for the rigorous accumulation of evidence for or against linkage allowing for both intra- and inter-sample heterogeneity. To date, the method has assumed linkage equilibrium between alleles at the trait locus and the marker locus. We now generalize the PPL to allow for linkage disequilibrium (LD), by incorporating variable phase probabilities into the underlying linkage likelihood. This enables us to recover the marginal posterior density of the recombination fraction, integrating out nuisance parameters of the trait model, including the locus heterogeneity (admixture) parameter, as well as a vector of LD parameters. The marginal posterior density can then be updated across data subsets or new data as they become available, while allowing parameters of the trait model to vary between data sets. The method applies immediately to general pedigree structures and to markers with multiple alleles. In the case of SNPs, the likelihood is parameterized in terms of the standard single LD parameter D′; and it therefore affords a mechanism for estimation of D′ between the marker and the trait, again, without fixing the parameters of the trait model and allowing for updating across data sets. It is even possible to allow for a different associated allele in different populations, while accumulating information regarding the strength of LD. While a computationally efficient implementation for multi-allelic markers is still in progress, we have implemented a version of this new LD-PPL for SNPs and evaluated its performance in nuclear families. Our simulations show that LD-PPLs tend to be larger than PPLs (stronger evidence in favor of linkage/LD) with increased LD level, under a variety of generating models; while in the absence of linkage and LD, LD-PPLs tend to be smaller than PPLs (stronger evidence against linkage). The estimate of D′ also behaves well even in relatively small, heterogeneous samples.

© 2005 S. Karger AG, Basel


  

Key Words

  • Trait-marker linkage disequilibrium
  • Disequilibrium estimation
  • Posterior probability of linkage
  • Bayesian linkage analysis

References

  1. Vieland VJ: Bayesian linkage analysis, or: How I learned to stop worrying and love the posterior probability of linkage. Am J Hum Genet 1998;63:947–954.
  2. Wang K, Vieland VJ, Huang J: A Bayesian approach to replication of linkage findings. Genet Epidemiol 1999;17(suppl 1):S749–754.
  3. Wang K, Huang J, Vieland VJ: The consistency of the posterior probability of linkage. Ann Hum Genet 2000;64(Pt 6):533–553.
  4. Vieland VJ, Wang K, Huang J: Power to detect linkage based on multiple sets of data in the presence of locus heterogeneity: Comparative evaluation of model-based linkage methods for affected sib pair data. Hum Hered 2001;51:199–208.
  5. Logue MW, Vieland VJ, Goedken RJ, Crowe RR: Bayesian analysis of a previously published genome screen for panic disorder reveals new and compelling evidence for linkage to chromosome 7. Am J Med Genet 2003;121B:95–99.
  6. Morton NE: Sequential tests for the detection of linkage. Am J Hum Genet 1955;7:277–318.
  7. Ewens WJ, Shute NC: A resolution of the ascertainment sampling problem. I. Theory. Theor Popul Biol 1986;30:388–412.
  8. Vieland VJ, Hodge SE: The problem of ascertainment for linkage analysis. Am J Hum Genet 1996;58:1072–1084.
  9. Vieland VJ, Logue M: HLODs, trait models, and ascertainment: implications of admixture for parameter estimation and linkage detection. Hum Hered 2002;53:23–35.
  10. Clerget-Darpoux F: Bias of the estimated recombination fraction and lod score due to an association between a disease gene and a marker gene. Ann Hum Genet 1982;46:363–372.
  11. Goldgar DE, Fain PR: Genetic Analysis Workshop II: Results of incorporating a linkage disequilibrium parameter. Genet Epidemiol 1984;1:179–182.
  12. Hellsten E, Vesa J, Speer MC, Makela TP, Jarvela I, Alitalo K, Ott J, Peltonen L: Refined assignment of the infantile neuronal ceroid lipofuscinosis (INCL, CLN1) locus at 1p32: Incorporation of linkage disequilibrium in multipoint analysis. Genomics 1983;16:720–725.
  13. Hastbacka J, de la Chapelle A, Mahtani MM, Clines G, Reeve-Daly MP, Daly M, Hamilton BA, Kusumi K, Trivedi B, Weaver A, et al: The diastrophic dysplasia gene encodes a novel sulfate transporter: Positional cloning by fine-structure linkage disequilibrium mapping. Cell 1994;78:1073–1087.
  14. Pratiwi R, Fletcher LM, Pyper WR, Do KA, Crawford DH, Powell LW, Jazwinska EC: Linkage disequilibrium analysis in Australian haemochromatosis patients indicates bipartite association with clinical expression. J Hepatol 1999;31:39–46.
  15. Annunen S, Paassilta P, Lohiniva J, Perala M, Pihlajamaa T, Karppinen J, Tervonen O, Kroger H, Lahde S, Vanharanta H, Ryhanen L, Göring HH, Ott J, Prockop DJ, Ala-Kokko L: An allele of COL9A2 associated with intervertebral disc disease. Science 1999;285:409–412.
  16. Slager SL, Huang J, Vieland VJ: Power comparisons between the TDT and two likelihood-based methods. Genet Epidemiol 2001;20:192–209.
  17. Huang J, Jiang Y: Linkage detection adaptive to linkage disequilibrium: The disequilibrium maximum-likelihood-binomial test for affected-sibship data. Am J Hum Genet 1999;65:1741–1759.
  18. Huang J, Jiang Y: The score statistic of the LD-lod analysis: Detecting linkage adaptive to linkage disequilibrium. Hum Hered 2001;52:83–98.
  19. Terwilliger JD, Göring HH: Gene mapping in the 20th and 21st centuries: Statistical methods, data analysis, and experimental design. Hum Biol 2000;72:63–132.
  20. Göring HH, Terwilliger JD: Linkage analysis in the presence of errors IV: Joint pseudomarker analysis of linkage and/or linkage disequilibrium on a mixture of pedigrees and singletons when the mode of inheritance cannot be accurately specified. Am J Hum Genet 2000;66:1310–1327.
  21. Smith CAB: Testing for heterogeneity of recombination fraction values in human genetics. Ann Hum Genet 1963;27:175–182.
  22. Logue MW, Vieland VJ: A new method for computing the multipoint posterior probability of linkage. Hum Hered 2004;57:90–99.
  23. Elston RC: Man bites dog? The validity of maximizing lod score to determine mode of inheritance. Am J Med Genet 1989;34:487–488.
  24. Ott J: Linkage analysis and family classification under heterogeneity. Ann Hum Genet 1983;47(Pt 4):311–320.
  25. Hedrick PW: Gametic disequilibrium measures: proceed with caution. Genetics 1987;117:331–341.
  26. Lewontin RC: The interaction of selection and linkage. I. General considerations: heterotic models. Genetics 1964;49:49–67.

    External Resources

  27. Hodge SE, Vieland VJ, Greenberg DA: HLODs remain powerful tools for detection of linkage in the presence of genetic heterogeneity. Am J Hum Genet 2002;70:556–557.
  28. Elston RC, Lange K: The prior probability of autosomal linkage. Ann Hum Genet 1975;38:341–350.
  29. Wang K: A Bayesian approach to replication of linage studies. Doctotal Thesis. The University of Iowa 1999.
  30. Vieland VJ, Ludington EA, Wang K, Huang J: The posterior probability of linkage (PPL) incorporating prior genetic information is efficient for detection of linkage and estimation of male/female recombination fraction rates for complex diseases. Am J Hum Genet 2000;67(suppl 2):328.
  31. Ott J: A computer program for linkage analysis of general human pedigrees. Am J Hum Genet 1976;28:528–529.
  32. Cavalli-Sforza LL, Bodmer WF: The Genetics of Human Populations San Francisco: WH Freeman 1971, pp 310–313.
  33. Royall R: Statistical Evidence: A Likelihood Paradigm. Chapman & Hall/CRC 1997.
  34. Govil M, Segre AM, Logue MW, Vieland VJ: MLIP: Parallel computation of LOD scores enabling full exploration of the trait-parameter space. Am J Hum Genetics 2003;S73:615.

  

Author Contacts

Dr. Veronica J. Vieland
Center for Statistical Genetics Research
2190 Westlawn, The University of Iowa
Iowa City, IA 52242 (USA)
Tel. +1 319 335 6722, Fax +1 319 353 3038, E-Mail veronica-vieland@uiowa.edu

  

Article Information

Received: March 3, 2005
Accepted after revision: April 29, 2005
Published online: July 7, 2005
Number of Print Pages : 10
Number of Figures : 2, Number of Tables : 8, Number of References : 34

  

Publication Details

Human Heredity (International Journal of Human and Medical Genetics)

Vol. 59, No. 4, Year 2005 (Cover Date: Released August 2005)

Journal Editor: Devoto, M. (Wilmington, Del.)
ISSN: 0001–5652 (print), 1423–0062 (Online)

For additional information: http://www.karger.com/hhe


Copyright / Drug Dosage / Disclaimer

Copyright: All rights reserved. No part of this publication may be translated into other languages, reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, microcopying, or by any information storage and retrieval system, without permission in writing from the publisher or, in the case of photocopying, direct payment of a specified fee to the Copyright Clearance Center.
Drug Dosage: The authors and the publisher have exerted every effort to ensure that drug selection and dosage set forth in this text are in accord with current recommendations and practice at the time of publication. However, in view of ongoing research, changes in goverment regulations, and the constant flow of information relating to drug therapy and drug reactions, the reader is urged to check the package insert for each drug for any changes in indications and dosage and for added warnings and precautions. This is particularly important when the recommended agent is a new and/or infrequently employed drug.
Disclaimer: The statements, opinions and data contained in this publication are solely those of the individual authors and contributors and not of the publishers and the editor(s). The appearance of advertisements or/and product references in the publication is not a warranty, endorsement, or approval of the products or services advertised or of their effectiveness, quality or safety. The publisher and the editor(s) disclaim responsibility for any injury to persons or property resulting from any ideas, methods, instructions or products referred to in the content or advertisements.

Abstract

The posterior probability of linkage (PPL) statistic has been developed as a method for the rigorous accumulation of evidence for or against linkage allowing for both intra- and inter-sample heterogeneity. To date, the method has assumed linkage equilibrium between alleles at the trait locus and the marker locus. We now generalize the PPL to allow for linkage disequilibrium (LD), by incorporating variable phase probabilities into the underlying linkage likelihood. This enables us to recover the marginal posterior density of the recombination fraction, integrating out nuisance parameters of the trait model, including the locus heterogeneity (admixture) parameter, as well as a vector of LD parameters. The marginal posterior density can then be updated across data subsets or new data as they become available, while allowing parameters of the trait model to vary between data sets. The method applies immediately to general pedigree structures and to markers with multiple alleles. In the case of SNPs, the likelihood is parameterized in terms of the standard single LD parameter D′; and it therefore affords a mechanism for estimation of D′ between the marker and the trait, again, without fixing the parameters of the trait model and allowing for updating across data sets. It is even possible to allow for a different associated allele in different populations, while accumulating information regarding the strength of LD. While a computationally efficient implementation for multi-allelic markers is still in progress, we have implemented a version of this new LD-PPL for SNPs and evaluated its performance in nuclear families. Our simulations show that LD-PPLs tend to be larger than PPLs (stronger evidence in favor of linkage/LD) with increased LD level, under a variety of generating models; while in the absence of linkage and LD, LD-PPLs tend to be smaller than PPLs (stronger evidence against linkage). The estimate of D′ also behaves well even in relatively small, heterogeneous samples.

© 2005 S. Karger AG, Basel


  

Author Contacts

Dr. Veronica J. Vieland
Center for Statistical Genetics Research
2190 Westlawn, The University of Iowa
Iowa City, IA 52242 (USA)
Tel. +1 319 335 6722, Fax +1 319 353 3038, E-Mail veronica-vieland@uiowa.edu

  

Article Information

Received: March 3, 2005
Accepted after revision: April 29, 2005
Published online: July 7, 2005
Number of Print Pages : 10
Number of Figures : 2, Number of Tables : 8, Number of References : 34

  

Publication Details

Human Heredity (International Journal of Human and Medical Genetics)

Vol. 59, No. 4, Year 2005 (Cover Date: Released August 2005)

Journal Editor: Devoto, M. (Wilmington, Del.)
ISSN: 0001–5652 (print), 1423–0062 (Online)

For additional information: http://www.karger.com/hhe


Article / Publication Details

First-Page Preview
Abstract of Original Paper

Received: 3/3/2005
Accepted: 4/29/2005
Published online: 8/19/2005
Issue release date: August 2005

Number of Print Pages: 10
Number of Figures: 2
Number of Tables: 8

ISSN: 0001-5652 (Print)
eISSN: 1423-0062 (Online)

For additional information: http://www.karger.com/HHE


Copyright / Drug Dosage

Copyright: All rights reserved. No part of this publication may be translated into other languages, reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, microcopying, or by any information storage and retrieval system, without permission in writing from the publisher or, in the case of photocopying, direct payment of a specified fee to the Copyright Clearance Center.
Drug Dosage: The authors and the publisher have exerted every effort to ensure that drug selection and dosage set forth in this text are in accord with current recommendations and practice at the time of publication. However, in view of ongoing research, changes in goverment regulations, and the constant flow of information relating to drug therapy and drug reactions, the reader is urged to check the package insert for each drug for any changes in indications and dosage and for added warnings and precautions. This is particularly important when the recommended agent is a new and/or infrequently employed drug.
Disclaimer: The statements, opinions and data contained in this publication are solely those of the individual authors and contributors and not of the publishers and the editor(s). The appearance of advertisements or/and product references in the publication is not a warranty, endorsement, or approval of the products or services advertised or of their effectiveness, quality or safety. The publisher and the editor(s) disclaim responsibility for any injury to persons or property resulting from any ideas, methods, instructions or products referred to in the content or advertisements.

References

  1. Vieland VJ: Bayesian linkage analysis, or: How I learned to stop worrying and love the posterior probability of linkage. Am J Hum Genet 1998;63:947–954.
  2. Wang K, Vieland VJ, Huang J: A Bayesian approach to replication of linkage findings. Genet Epidemiol 1999;17(suppl 1):S749–754.
  3. Wang K, Huang J, Vieland VJ: The consistency of the posterior probability of linkage. Ann Hum Genet 2000;64(Pt 6):533–553.
  4. Vieland VJ, Wang K, Huang J: Power to detect linkage based on multiple sets of data in the presence of locus heterogeneity: Comparative evaluation of model-based linkage methods for affected sib pair data. Hum Hered 2001;51:199–208.
  5. Logue MW, Vieland VJ, Goedken RJ, Crowe RR: Bayesian analysis of a previously published genome screen for panic disorder reveals new and compelling evidence for linkage to chromosome 7. Am J Med Genet 2003;121B:95–99.
  6. Morton NE: Sequential tests for the detection of linkage. Am J Hum Genet 1955;7:277–318.
  7. Ewens WJ, Shute NC: A resolution of the ascertainment sampling problem. I. Theory. Theor Popul Biol 1986;30:388–412.
  8. Vieland VJ, Hodge SE: The problem of ascertainment for linkage analysis. Am J Hum Genet 1996;58:1072–1084.
  9. Vieland VJ, Logue M: HLODs, trait models, and ascertainment: implications of admixture for parameter estimation and linkage detection. Hum Hered 2002;53:23–35.
  10. Clerget-Darpoux F: Bias of the estimated recombination fraction and lod score due to an association between a disease gene and a marker gene. Ann Hum Genet 1982;46:363–372.
  11. Goldgar DE, Fain PR: Genetic Analysis Workshop II: Results of incorporating a linkage disequilibrium parameter. Genet Epidemiol 1984;1:179–182.
  12. Hellsten E, Vesa J, Speer MC, Makela TP, Jarvela I, Alitalo K, Ott J, Peltonen L: Refined assignment of the infantile neuronal ceroid lipofuscinosis (INCL, CLN1) locus at 1p32: Incorporation of linkage disequilibrium in multipoint analysis. Genomics 1983;16:720–725.
  13. Hastbacka J, de la Chapelle A, Mahtani MM, Clines G, Reeve-Daly MP, Daly M, Hamilton BA, Kusumi K, Trivedi B, Weaver A, et al: The diastrophic dysplasia gene encodes a novel sulfate transporter: Positional cloning by fine-structure linkage disequilibrium mapping. Cell 1994;78:1073–1087.
  14. Pratiwi R, Fletcher LM, Pyper WR, Do KA, Crawford DH, Powell LW, Jazwinska EC: Linkage disequilibrium analysis in Australian haemochromatosis patients indicates bipartite association with clinical expression. J Hepatol 1999;31:39–46.
  15. Annunen S, Paassilta P, Lohiniva J, Perala M, Pihlajamaa T, Karppinen J, Tervonen O, Kroger H, Lahde S, Vanharanta H, Ryhanen L, Göring HH, Ott J, Prockop DJ, Ala-Kokko L: An allele of COL9A2 associated with intervertebral disc disease. Science 1999;285:409–412.
  16. Slager SL, Huang J, Vieland VJ: Power comparisons between the TDT and two likelihood-based methods. Genet Epidemiol 2001;20:192–209.
  17. Huang J, Jiang Y: Linkage detection adaptive to linkage disequilibrium: The disequilibrium maximum-likelihood-binomial test for affected-sibship data. Am J Hum Genet 1999;65:1741–1759.
  18. Huang J, Jiang Y: The score statistic of the LD-lod analysis: Detecting linkage adaptive to linkage disequilibrium. Hum Hered 2001;52:83–98.
  19. Terwilliger JD, Göring HH: Gene mapping in the 20th and 21st centuries: Statistical methods, data analysis, and experimental design. Hum Biol 2000;72:63–132.
  20. Göring HH, Terwilliger JD: Linkage analysis in the presence of errors IV: Joint pseudomarker analysis of linkage and/or linkage disequilibrium on a mixture of pedigrees and singletons when the mode of inheritance cannot be accurately specified. Am J Hum Genet 2000;66:1310–1327.
  21. Smith CAB: Testing for heterogeneity of recombination fraction values in human genetics. Ann Hum Genet 1963;27:175–182.
  22. Logue MW, Vieland VJ: A new method for computing the multipoint posterior probability of linkage. Hum Hered 2004;57:90–99.
  23. Elston RC: Man bites dog? The validity of maximizing lod score to determine mode of inheritance. Am J Med Genet 1989;34:487–488.
  24. Ott J: Linkage analysis and family classification under heterogeneity. Ann Hum Genet 1983;47(Pt 4):311–320.
  25. Hedrick PW: Gametic disequilibrium measures: proceed with caution. Genetics 1987;117:331–341.
  26. Lewontin RC: The interaction of selection and linkage. I. General considerations: heterotic models. Genetics 1964;49:49–67.

    External Resources

  27. Hodge SE, Vieland VJ, Greenberg DA: HLODs remain powerful tools for detection of linkage in the presence of genetic heterogeneity. Am J Hum Genet 2002;70:556–557.
  28. Elston RC, Lange K: The prior probability of autosomal linkage. Ann Hum Genet 1975;38:341–350.
  29. Wang K: A Bayesian approach to replication of linage studies. Doctotal Thesis. The University of Iowa 1999.
  30. Vieland VJ, Ludington EA, Wang K, Huang J: The posterior probability of linkage (PPL) incorporating prior genetic information is efficient for detection of linkage and estimation of male/female recombination fraction rates for complex diseases. Am J Hum Genet 2000;67(suppl 2):328.
  31. Ott J: A computer program for linkage analysis of general human pedigrees. Am J Hum Genet 1976;28:528–529.
  32. Cavalli-Sforza LL, Bodmer WF: The Genetics of Human Populations San Francisco: WH Freeman 1971, pp 310–313.
  33. Royall R: Statistical Evidence: A Likelihood Paradigm. Chapman & Hall/CRC 1997.
  34. Govil M, Segre AM, Logue MW, Vieland VJ: MLIP: Parallel computation of LOD scores enabling full exploration of the trait-parameter space. Am J Hum Genetics 2003;S73:615.