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Table of Contents
Vol. 50, No. 6, 2000
Issue release date: November–December 2000 (July 2000)
Section title: Original Paper
Hum Hered 2000;50:334–349
(DOI:10.1159/000022939)

A Complete Enumeration and Classification of Two-Locus Disease Models

Li W.a · Reich J.b
aLaboratory of Statistical Genetics, Rockefeller University, New York, N.Y., USA and bDepartment of Biomathematics, Max-Delbrück-Centrum, Berlin-Buch, Germany
email Corresponding Author

Abstract

There are 512 two-locus, two-allele, two-phenotype, fully penetrant disease models. Using the permutation between two alleles, between two loci, and between being affected and unaffected, one model can be considered to be equivalent to another model under the corresponding permutation. These permutations greatly reduce the number of two-locus models in the analysis of complex diseases. This paper determines the number of nonredundant two-locus models (which can be 102, 100, 96, 51, 50, or 58, depending on which permutations are used, and depending on whether zero-locus and single-locus models are excluded). Whenever possible, these nonredundant two-locus models are classified by their property. Besides the familiar features of multiplicative models (logical AND), heterogeneity models (logical OR), and threshold models, new classifications are added or expanded: modifying-effect models, logical XOR models, interference and negative interference models (neither dominant nor recessive), conditionally dominant/recessive models, missing lethal genotype models, and highly symmetric models. The following aspects of two-locus models are studied: the marginal penetrance tables at both loci, the expected joint identity-by-descent (IBD) probabilities, and the correlation between marginal IBD probabilities at the two loci. These studies are useful for linkage analyses using single-locus models while the underlying disease model is two-locus, and for correlation analyses using the linkage signals at different locations obtained by a single-locus model.

© 2000 S. Karger AG, Basel


  

Key Words

  • Two-locus model
  • Epistasis
  • Identity by descent
  • Correlation

References

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  3. Bishop CM: Neural Networks for Pattern Recognition. Oxford, Oxford University Press, 1995.
  4. Clerget-Darpoux F, Babron MC: Testing genetic models for IDDM by the MASC method. Genet Epidemiol 1989;6:59–64.
  5. Cotterman CW: Regular two-allele and three-allele phenotype systems. Am J Hum Genet 1953;5:193–235.
  6. Defrise-Gussenhoven PE: Hypothèses de dimérie et de non-pénétrance. Acta Genet Stat Med (Basel) 1961;12:65–69.
  7. Campbell MA, Elston RC: Relatives of probands: Models for preliminary genetic analysis. Ann Hum Genet 1971;35:225–236.

    External Resources

  8. Cockerham CC: An extension of the concept of partitioning hereditary variance for analysis of covariances among relatives when epistasis is present. Genetics 1954;39:859–882.
  9. Cordell HJ, Todd JA, Bennett ST, Kawaguchi Y, Farrall M: Two-locus maximum lod score analysis of a multifactorial trait: Joint consideration of IDDM2 and IDDM4 with IDDM1 in type I diabetes. Am J Hum Genet 1995;57:920–934.

    External Resources

  10. Cox NJ, Frigge M, Nicolae DL, Concannon P, Hanis CL, Bell GI, Kong A: Loci on chromosomes 2 (NIDDM1) and 15 interact to increase susceptibility diabetes in Mexican Americans. Nat Genet 1999;21:213–215.
  11. Davis S, Schroeder M, Goldin LR, Weeks DE: Nonparametric simulation-based statistics for detecting linkage in general pedigrees. Am J Hum Genet 1996;58:867–880.

    External Resources

  12. Davis S, Weeks DE: Comparison of nonparametric statistics for detection of linkage in nuclear families: Single-marker evaluation. Am J Hum Genet 1997;61:1431–1444.
  13. de Bruijn NG: Pólya theory of counting; in Beckenbach EF (ed): Applied Combinatorial Mathematics. New York, Wiley, 1964, pp 144–184.
  14. Dizier MH, Babron MC, Clerget-Darpoux F: Interactive effect of two candidate genes in a disease: Extension of the marker-association-segregation χ2 method. Am J Hum Genet 1994;55:1042–1049.
  15. Dizier MH, Babron MC, Clerget-Darpoux F: Conclusion of LOD-score analysis for family data generated under two-locus models. Am J Hum Genet 1996;58:1338–1346.

    External Resources

  16. Dizier MH, Bonaiti-Pellie C, Clerget-Darpoux F: Conclusions of segregation analysis for family data generated under two-locus models. Am J Hum Genet 1993;53:1338–1346.
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    External Resources

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  32. Greenberg DA: Simulation studies of segregation analysis: Application to two-locus models. Am J Hum Genet 1984;36:167–176.
  33. Greenberg DA: Linkage analysis assuming a single-locus mode of inheritance for traits determined by two loci: Inferring mode of inheritance and estimating penetrance. Genet Epidemiol 1990;7:467–479.
  34. Greenberg DA, Lange KL: A maximum likelihood test of the two locus model for coeliac disease. Am J Med Genet 1982;12:75–82.
  35. Greenberg DA, Delgado-Escueta AV, Maldonado HM, Widelitz H: Segregation analysis of juvenile myoclonic epilepsy. Genet Epidemiol 1988;5:81–94.

    External Resources

  36. Greenberg DA, Hodge SE: Linkage analysis under ‘random’ and ‘genetic’ reduced penetrance. Genet Epidemiol 1989;6:259–264.
  37. Greenberg DA, Abreu P, Hodge SE: The power to detect linkage in complex disease by means of simple LOD-score analyses. Am J Hum Genet 1998;63:870–879.
  38. Hartl DL, Maruyama T: Phenogram enumeration: The number of regular genotype-phenotype correspondences in genetic systems. J Theor Biol 1968;20:129–163.

    External Resources

  39. Hodge SE: Some epistatic two-locus models of disease. I. Relative risks and identical-by-descent distributions in affected sib pairs. Am J Hum Genet 1981;33:381–395.
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Author Contacts

Wentian Li
Laboratory of Statistical Genetics, Box 192
Rockefeller University, 1230 York Avenue
New York, NY 10021 (USA)
Tel. +1 212 327 7977, Fax +1 212 327 7996, E-Mail wli@linkage.rockefeller.edu

  

Article Information

Received: Received: April 7, 1999
Revision received: July 23, 1999
Accepted: August 5, 1999
Number of Print Pages : 16
Number of Figures : 0, Number of Tables : 7, Number of References : 94

  

Publication Details

Human Heredity (International Journal of Human and Medical Genetics)
Founded 1950 as Acta Genetica et Statistica Medica by Gunnar Dahlberg; Continued by M. Hauge (1965–1983)

Vol. 50, No. 6, Year 2000 (Cover Date: November-December 2000 (Released July 2000))

Journal Editor: J. Ott, New York, N.Y.
ISSN: 0001–5652 (print), 1423–0062 (Online)

For additional information: http://www.karger.ch/journals/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

There are 512 two-locus, two-allele, two-phenotype, fully penetrant disease models. Using the permutation between two alleles, between two loci, and between being affected and unaffected, one model can be considered to be equivalent to another model under the corresponding permutation. These permutations greatly reduce the number of two-locus models in the analysis of complex diseases. This paper determines the number of nonredundant two-locus models (which can be 102, 100, 96, 51, 50, or 58, depending on which permutations are used, and depending on whether zero-locus and single-locus models are excluded). Whenever possible, these nonredundant two-locus models are classified by their property. Besides the familiar features of multiplicative models (logical AND), heterogeneity models (logical OR), and threshold models, new classifications are added or expanded: modifying-effect models, logical XOR models, interference and negative interference models (neither dominant nor recessive), conditionally dominant/recessive models, missing lethal genotype models, and highly symmetric models. The following aspects of two-locus models are studied: the marginal penetrance tables at both loci, the expected joint identity-by-descent (IBD) probabilities, and the correlation between marginal IBD probabilities at the two loci. These studies are useful for linkage analyses using single-locus models while the underlying disease model is two-locus, and for correlation analyses using the linkage signals at different locations obtained by a single-locus model.

© 2000 S. Karger AG, Basel


  

Author Contacts

Wentian Li
Laboratory of Statistical Genetics, Box 192
Rockefeller University, 1230 York Avenue
New York, NY 10021 (USA)
Tel. +1 212 327 7977, Fax +1 212 327 7996, E-Mail wli@linkage.rockefeller.edu

  

Article Information

Received: Received: April 7, 1999
Revision received: July 23, 1999
Accepted: August 5, 1999
Number of Print Pages : 16
Number of Figures : 0, Number of Tables : 7, Number of References : 94

  

Publication Details

Human Heredity (International Journal of Human and Medical Genetics)
Founded 1950 as Acta Genetica et Statistica Medica by Gunnar Dahlberg; Continued by M. Hauge (1965–1983)

Vol. 50, No. 6, Year 2000 (Cover Date: November-December 2000 (Released July 2000))

Journal Editor: J. Ott, New York, N.Y.
ISSN: 0001–5652 (print), 1423–0062 (Online)

For additional information: http://www.karger.ch/journals/hhe


Article / Publication Details

First-Page Preview
Abstract of Original Paper

Published online: 7/19/2000
Issue release date: November–December 2000 (July 2000)

Number of Print Pages: 16
Number of Figures: 0
Number of Tables: 7

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. Agresti A: An Introduction to Categorical Data Analysis. New York, Wiley, 1996.
  2. Badner JA, Gershon ES, Goldin LR: Optimal ascertainment strategies to detect linkage to common disease alleles. Am J Hum Genet 1998;63:880–888.
  3. Bishop CM: Neural Networks for Pattern Recognition. Oxford, Oxford University Press, 1995.
  4. Clerget-Darpoux F, Babron MC: Testing genetic models for IDDM by the MASC method. Genet Epidemiol 1989;6:59–64.
  5. Cotterman CW: Regular two-allele and three-allele phenotype systems. Am J Hum Genet 1953;5:193–235.
  6. Defrise-Gussenhoven PE: Hypothèses de dimérie et de non-pénétrance. Acta Genet Stat Med (Basel) 1961;12:65–69.
  7. Campbell MA, Elston RC: Relatives of probands: Models for preliminary genetic analysis. Ann Hum Genet 1971;35:225–236.

    External Resources

  8. Cockerham CC: An extension of the concept of partitioning hereditary variance for analysis of covariances among relatives when epistasis is present. Genetics 1954;39:859–882.
  9. Cordell HJ, Todd JA, Bennett ST, Kawaguchi Y, Farrall M: Two-locus maximum lod score analysis of a multifactorial trait: Joint consideration of IDDM2 and IDDM4 with IDDM1 in type I diabetes. Am J Hum Genet 1995;57:920–934.

    External Resources

  10. Cox NJ, Frigge M, Nicolae DL, Concannon P, Hanis CL, Bell GI, Kong A: Loci on chromosomes 2 (NIDDM1) and 15 interact to increase susceptibility diabetes in Mexican Americans. Nat Genet 1999;21:213–215.
  11. Davis S, Schroeder M, Goldin LR, Weeks DE: Nonparametric simulation-based statistics for detecting linkage in general pedigrees. Am J Hum Genet 1996;58:867–880.

    External Resources

  12. Davis S, Weeks DE: Comparison of nonparametric statistics for detection of linkage in nuclear families: Single-marker evaluation. Am J Hum Genet 1997;61:1431–1444.
  13. de Bruijn NG: Pólya theory of counting; in Beckenbach EF (ed): Applied Combinatorial Mathematics. New York, Wiley, 1964, pp 144–184.
  14. Dizier MH, Babron MC, Clerget-Darpoux F: Interactive effect of two candidate genes in a disease: Extension of the marker-association-segregation χ2 method. Am J Hum Genet 1994;55:1042–1049.
  15. Dizier MH, Babron MC, Clerget-Darpoux F: Conclusion of LOD-score analysis for family data generated under two-locus models. Am J Hum Genet 1996;58:1338–1346.

    External Resources

  16. Dizier MH, Bonaiti-Pellie C, Clerget-Darpoux F: Conclusions of segregation analysis for family data generated under two-locus models. Am J Hum Genet 1993;53:1338–1346.
  17. Dizier MH, Clerget-Darpoux F: Two-disease locus model: Sib pair method using information on both HLA and Gm. Genet Epidemiol 1986;3:343–356.

    External Resources

  18. Dizier MH, Clerget-Darpoux F, Hochez J: Segregation analysis of two genetic markers in IDDM families under two-locus models. Genet Epidemiol 1989;6:71–75.
  19. Dizier MH, Clerget-Darpoux F, Hochez J: Two-disease-locus model: Segregation analysis using information on two markers in nuclear families. Application to IDDM. Tissue Antigens 1990;36:1–7.
  20. Durner M, Greenberg DA, Hodge SE: Inter- and intra-familial heterogeneity: Effective sampling strategies and comparison of analysis methods. Am J Hum Genet 1992;51:859–870.
  21. Durner M, Vieland VJ, Greenberg DA: Further evidence for the increased power of LOD scores compared with nonparametric methods. Am J Hum Genet 1999;64:281–289.
  22. Eccles DM, Forabosco P, Williams A, Dunn B, Williams C, Bishop DT, Morton NE: Segregation analysis of ovarian cancer using diathesis to include other cancers. Am J Hum Genet 1997;61:243–252.
  23. Falk CT: Effect of genetic heterogeneity and assortative mating on linkage analysis: A simulation study. Am J Hum Genet 1997;61:1169–1178.
  24. Farrall M: Affected sibpair linkage tests for multiple linked susceptibility genes. Genet Epidemiol 1997;14:103–115.
  25. Fisher RA: The correlation between relatives on the supposition of Mendelian inheritance. Trans R Soc Edinburgh 1918;52:399–433.
  26. Frankel WN, Schork NJ: Who’s afraid of epistasis? Nat Genet 1996;14:371–373.
  27. Goldgar DE, Oniki RS: Comparison of a multipoint identity-by-descent method with parametric multipoint linkage analysis for mapping quantitative traits. Am J Hum Genet 1992;50:598–606.
  28. Goldgar DE, Easton DF: Optimal strategies for mapping complex diseases in the presence of multiple loci. Am J Hum Genet 1997;60:1222–1232.
  29. Goldin LR: Detection of linkage under heterogeneity: Comparison of the two-locus vs. admixture models. Genet Epidemiol 1992;9:61–66.
  30. Goldin LR, Weeks DE: Two-locus models of disease: Comparison of likelihood and nonparametric linkage methods. Am J Hum Genet 1993;53:908–915.
  31. Greenberg DA: A simple method for testing two-locus models of inheritance. Am J Hum Genet 1981;33:519–530.
  32. Greenberg DA: Simulation studies of segregation analysis: Application to two-locus models. Am J Hum Genet 1984;36:167–176.
  33. Greenberg DA: Linkage analysis assuming a single-locus mode of inheritance for traits determined by two loci: Inferring mode of inheritance and estimating penetrance. Genet Epidemiol 1990;7:467–479.
  34. Greenberg DA, Lange KL: A maximum likelihood test of the two locus model for coeliac disease. Am J Med Genet 1982;12:75–82.
  35. Greenberg DA, Delgado-Escueta AV, Maldonado HM, Widelitz H: Segregation analysis of juvenile myoclonic epilepsy. Genet Epidemiol 1988;5:81–94.

    External Resources

  36. Greenberg DA, Hodge SE: Linkage analysis under ‘random’ and ‘genetic’ reduced penetrance. Genet Epidemiol 1989;6:259–264.
  37. Greenberg DA, Abreu P, Hodge SE: The power to detect linkage in complex disease by means of simple LOD-score analyses. Am J Hum Genet 1998;63:870–879.
  38. Hartl DL, Maruyama T: Phenogram enumeration: The number of regular genotype-phenotype correspondences in genetic systems. J Theor Biol 1968;20:129–163.

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