Skip to main content
SpringerLink
Log in

Simple linear approximations to the likelihood equation for combining evidence in multiple 2×2 tables: A critique of conventional procedures

  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

Summary

The conventional procedures for a common odds ratio in multiple 2×2 tables are explored and critiqued. Three types of linear approximation to the likelihood equations under some models of common measures of association are used to derive the popular conventional estimators and test statistics. Some of them are derived using the model of the common standardized difference which is an unacceptable measure. The derivation provides us with some characteristics of the procedures. The advantages of procedures based on the conditional and unconditional likelihoods are discussed.

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

Similar content being viewed by others

References

  1. Andersen, E. B. (1970). Asymptotic properties of conditional maximum-likelihood estimators,J. R. Statist. Soc., B,32, 283–301.

    MathSciNet  MATH  Google Scholar 

  2. Anscombe, F. J. (1956). On estimating binomial response relations,Biometrika,43, 461–464.

    Article  MathSciNet  Google Scholar 

  3. Birch, M. W. (1964). The detection of partial association I: The 2×2 case,J. R. Statist. Soc., B,26, 313–324.

    MathSciNet  MATH  Google Scholar 

  4. Breslow, N. E. (1981). Odds ratio estimators when the data are sparse,Biometrika,68, 73–84.

    Article  MathSciNet  Google Scholar 

  5. Breslow, N. E. and Liang, K. Y. (1982). The variance of the Mantel-Haenszel estimator,Biometrics,38, 943–952.

    Article  Google Scholar 

  6. Cochran, W. G. (1954). Some methods for strengthening the common χ2 tests,Biometrics,10, 417–451.

    Article  MathSciNet  Google Scholar 

  7. Cox, D. R. (1970).The Analysis of Binary Data, Methuen and Co., London.

    MATH  Google Scholar 

  8. Day, N. E. and Byar, D. P. (1979). Testing hypotheses in case-control studies—equivalence of Mantel-Haenszel statistics and logistic score tests,Biometrics,35, 623–630.

    Article  MathSciNet  Google Scholar 

  9. Farewell, V. T. and Prentice, R. L. (1980). The approximation of partial likelihood with emphasis on case-control studies,Biometrika,67, 273–278.

    Article  MathSciNet  Google Scholar 

  10. Fleiss, J. L. (1970). On the asserted invariance of the odds ratio,Brit. J. Prev. Soc. Med.,24, 45–46.

    Google Scholar 

  11. Fleiss, J. L. (1981).Statistical Methods for Rates and Proportions, 2nd ed., Wiley, New York.

    MATH  Google Scholar 

  12. Gart, J. J. (1971). The comparison of proportions: A review of significant tests, confidence intervals and adjustments for stratification,Rev. Int. Statist. Inst.,39, 148–169.

    Article  Google Scholar 

  13. Goodman, L. A. (1969). On partitioning χ2 and detecting partial association in three-way contingency tables,J. R. Statist. Soc. B,31, 486–498.

    MathSciNet  MATH  Google Scholar 

  14. Haldane, J. B. S. (1955). The estimation and significance of the logarithm of a ratio of frequencies,Ann. Hum. Genet.,20, 309–311.

    Article  Google Scholar 

  15. Halperin, M. et al. (1977). Testing for interaction in anI×J×K contingency table,Biometrika,64, 271–275.

    MathSciNet  Google Scholar 

  16. Harkness, W. L. (1965). Properties of the extended hypergeometric distribution,Ann. Math. Statist.,36, 938–945.

    Article  MathSciNet  Google Scholar 

  17. Hauck, W. W. (1979). The large sample variance of the Mantel-Haenszel estimator of a common odds ratio,Biometrics,35, 817–819.

    Article  MathSciNet  Google Scholar 

  18. Hauck, W. W., Anderson, S. and Leahy, F. L. (1982). Finite-sample properties of some old and some new estimators of a common odds ratio from multiple 2×2 tables,J. Amer. Statist. Ass.,77, 145–152.

    MATH  Google Scholar 

  19. Hitchcock, S. E. (1962). A note on the estimation of the parameters of the logistic function, using the minimum logit χ2 method,Biometrika,49, 250–255.

    MATH  Google Scholar 

  20. Lubin, J. H. (1981). An empirical evaluation of the use of conditional and unconditional likelihoods for case-control data,Biometrika,68, 567–571.

    Article  Google Scholar 

  21. Mantel, N. and Haenszel, W. (1959). Statistical aspects of the analysis of data from retrospective studies of disease,J. Nat. Cancer Inst.,22, 719–748.

    Google Scholar 

  22. Mantel, N., Brown, C. and Byar, D. D. (1977). Tests for homogeneity of effect in an epidemiologic investigation,Amer. J. Epidemiol.,106, 125–129.

    Article  Google Scholar 

  23. Mckinlay, S. M. (1975). The effect of bias on estimators of relative risk for pairmatched and stratified samples,J. Amer. Statist. Ass.,70, 859–864.

    Article  Google Scholar 

  24. Mckinlay, S. M. (1978). The effect of nonzero second-order interaction on combined estimators of the odds ratio,Biometrika,65, 191–202.

    Article  Google Scholar 

  25. Nelder, J. A. and Wedderburn, R. W. M. (1972). Generalized linear models,J. R. Statist. Soc., A,135, 370–384.

    Article  Google Scholar 

  26. Radhakrishna, S. (1965). Combination of results from several 2×2 contingency tables,Biometrics,21, 86–98.

    Article  MathSciNet  Google Scholar 

  27. Woolf, B. (1955). On estimating the relationship between blood group and disease,Ann. Hum. Genet.,19, 251–253.

    Article  Google Scholar 

  28. Yanagimoto, T. and Kamakura, T. (1984). The maximum full and partial likelihood estimators in the proportional hazard models,Ann. Inst. Statist. Math., A,36, 363–373.

    Article  MathSciNet  Google Scholar 

  29. Yates, F. (1955). The use of transformations and maximum likelihood in the analysis of quantal experiments involving two treatments,Biometrika,42, 382–411.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Okayama University of Science, Okayama, Japan

    Takemi Yanagimoto & Eiji Yamamoto

Authors
  1. Takemi Yanagimoto
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eiji Yamamoto
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

The Institute of Statistical Mathematics

About this article

Cite this article

Yanagimoto, T., Yamamoto, E. Simple linear approximations to the likelihood equation for combining evidence in multiple 2×2 tables: A critique of conventional procedures. Ann Inst Stat Math 37, 37–49 (1985). https://doi.org/10.1007/BF02481079

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02481079

Key words and phrases

Search

Navigation