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Counting Methods and the EM Algorithm

  • Kenneth Lange
Part of the Statistics for Biology and Health book series (SBH)

Abstract

In this chapter and the next, we undertake the study of estimation methods and their applications in genetics. Because of the complexity of genetic models, geneticists by and large rely on maximum likelihood estimators rather than on competing estimators derived from minimax, invariance, robustness, or Bayesian principles. A host of methods exists for numerically computing maximum likelihood estimates. Some of the most appealing involve simple counting arguments and the EM algorithm. Indeed, historically geneticists devised many special cases of the EM algorithm before it was generally formulated by Dempster et al. [5,9]. Our initial example retraces some of the steps in the long march from concrete problems to an abstract algorithm applicable to an astonishing variety of statistical models.

Keywords

Success Probability Affected Sibling Color Blindness Codominant Allele Complete Data Likelihood 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Chakraborty R, Srinivasan MR, Daiger SP (1993) Evaluation of standard error and confidence interval of estimated multilocus genotype probabilities, and their applications in DNA forensics. Amer J Hum Genet 52:60–70Google Scholar
  2. [2]
    Clarke CA, Price-Evans DA, McConnell RB, Sheppard PM (1959) Secretion of blood group antigens and peptic ulcers. Brit Med J 1:603–607CrossRefGoogle Scholar
  3. [3]
    Crow JF (1965) Problems of ascertainment in the analysis of family data. Epidemiology and Genetics of Chronic Disease. Public Health Service Publication 1163, Neel JV, Shaw MW, Schull WJ, editors, Department of Health, Education, and Welfare, Washington, DCGoogle Scholar
  4. [4]
    Crow JF (1986) Basic Concepts in Population, Quantitative, and Ecological Genetics. Freeman, San FranciscoGoogle Scholar
  5. [5]
    Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm (with discussion). J Roy Stat Soc B 39:1–38zbMATHMathSciNetGoogle Scholar
  6. [6]
    Little RJA, Rubin DB (1987) Statistical Analysis with Missing Data. Wiley, New YorkzbMATHGoogle Scholar
  7. [7]
    Ott J (1977) Counting methods (EM algorithm) in human pedigree analysis: linkage and segregation analysis. Ann Hum Genet 40:443–454zbMATHGoogle Scholar
  8. [8]
    Rao CR (1975) Linear Statistical Inference and Its Applications, 2nd ed. Wiley, New YorkGoogle Scholar
  9. [9]
    Smith CAB (1957) Counting methods in genetical statistics. Ann Hum Genet 21:254–276CrossRefGoogle Scholar
  10. [10]
    Tanner MA (1993) Tools for Statistical Inference: Methods for the Exploration of Posterior Distributions and Likelihood Functions, 2nd ed. Springer-Verlag, New YorkzbMATHGoogle Scholar
  11. [11]
    Titterington DM, Smith AFM, Makov UE (1985) Statistical Analysis of Finite Mixture Distributions. Wiley, New YorkzbMATHGoogle Scholar
  12. [12]
    Weeks DE, Lange K (1989) Trials, tribulations, and triumphs of the EM algorithm in pedigree analysis. IMA J Math Appl Med Biol 6:209–232zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    Weir BS (1990) Genetic Data Analysis. Sinauer, Sunderland, MAGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Kenneth Lange
    • 1
  1. 1.Department of Biostatistics and MathematicsUniversity of MichiganAnn ArborUSA

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