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Biostatistics pp 125–149Cite as

Estimation of a Common Odds Ratio

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Part of the book series: The University of Western Ontario Series in Philosophy of Science ((WONS,volume 38))

Abstract

For comparing two proportions from stratified samples, one approach is via inference on the odds ratio. The various point and interval estimators of a common odds ratio from multiple 2 × 2 tables are reviewed in this paper, with particular emphasis on the point estimators, their asymptotic properties, and what is known about finite-sample properties. Based on research to date, the conditional maximum likelihood and Mantel-Haenszel estimators are recommended as the point estimators of choice. Confidence interval methods have not been studied as well, but there is a method associated with the Mantel-Haenszel estimator that is a good choice. There is the possibility of improvement in the finite-sample properties of these estimators. However, further work is needed before one of these modifications can be recommended for general use.

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References

  • Andersen, E. B. (1970), “Asymptotic properties of conditional maximum-likelihood estimators”. Journal of the Royal Statistical Society, Series B 32, 283–301.

    MATH  Google Scholar 

  • Andersen, E. B. (1973), Conditional Inference and Models for Measuring. Copen hagen: Mentalhygienisk Forlag.

    Google Scholar 

  • Anderson, S., A. Auquier, W. W., Hauck, D. Oakes, W. Vandaele, and H. Weisberg (1980), Statistical Methods for Comparative Studies. New York: Wiley and Sons.

    Book  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Berkson, J. (1944), “Application of the logit function to bioassay”. Journal of the American Statistical Association 39, 357–365.

    Article  Google Scholar 

  • Birch, M. W. (1964), “The detection of partial association, I: the 2 × 2 case”. Journal of the Royal Statisticai Society, Series B 26, 313–324.

    MathSciNet  MATH  Google Scholar 

  • Breslow, N. E. (1976), “Regression analysis of the log odds ratio: A method for retrospective studies”. Biometrics 32, 409–416.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Breslow, N. E., and N. E. Day (1980), Statisticai MetAods in Cancer Research, Volume 1—The Analysis of Case-Control Studies. Lyon: International Agency for Research on Cancer.

    Google Scholar 

  • Breslow, N. E., and K. Y. Liang (1982), “The variance of the Mantel-Haenszel estimator”. Biometrics 88, 943–952; Errata: Biometrics 40, 1217 (1984).

    Google Scholar 

  • Brown, C. C. (1981), “The validity of approximation methods for interval estimation of the odds ratio”. American Journal of Epidemiology 113, 474–480.

    Google Scholar 

  • Centers for Disease Control Cancer and Steroid Hormone Study (1983), “Longterm oral contraceptive use and the risk of breast cancer.” Journal of the American Medical Association 249, 1591–1595.

    Article  Google Scholar 

  • Connett, J., A. Ejigou, R. McHugh, and N. Breslow (1981), “The precision of the Mantel-Haenszel estimator in case-control studies with multiple matching”. American Journal of Epidemiology 116, 875–877.

    Google Scholar 

  • Cornfield, J. (1956), “A statistical problem arising from retrospective studies”. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability 4, 135–148.

    MathSciNet  Google Scholar 

  • Davis, L. J. (1985), “Weighted averages of the observed odds ratios when the number of tables is large”. Biometrika 72, 203–205.

    Article  MathSciNet  MATH  Google Scholar 

  • Donald, A. (1984), “The analysis of clustered data in sets of 2 × 2 contingency tables.” Ph.D. thesis, The University of Western Ontario.

    Google Scholar 

  • Ejigou, A., and R. McHugh (1981), “Relative risk estimation under multiple matching”. Biometrika 68, 85–91.

    Article  MathSciNet  MATH  Google Scholar 

  • Fienberg, S. E., and P. Holland (1970), “Methods for eliminating zero counts in contingency tables”. In Random Counts in Models and Structures, ed. G. P. Patil, pp. 233–260. University Park: Pennsylvania State University Press.

    Google Scholar 

  • Fisher, R. A. (1935), “The logic of inductive inference”. Journal of the Royal Statistical Society, Series A 98, 39–54.

    Article  Google Scholar 

  • Fleiss, J. L. (1979), “Confidence intervals for the odds ratio in case-control studies: the state of the art”. Journal of Chronic Diseases 32, 69–77.

    Article  Google Scholar 

  • Fleiss, J. L. (1984), “The Mantel-Haenszel estimator in case-control studies with varying numbers of controls matched to each case”. American Journal of Epi demiology 120, 1–3.

    Google Scholar 

  • Gart, J. J. (1962), “On the combination of relative risks”. Biometrics 18, 601–610.

    Article  MATH  Google Scholar 

  • Gart, J. J. (1966), “Alternative analyses of contingency tables”. Journal of the Royal Statistical Society, Series B 28, 164–179.

    MathSciNet  Google Scholar 

  • Gart, J. J. (1970), “Point and interval estimation of the common odds ratio in the combination of 2 × 2 tables with fixed marginals”. Biometrika 57, 471–475.

    MATH  Google Scholar 

  • Gart, J. J., (1971), “The comparison of proportions: a review of significance tests, confidence intervals and adjustments for stratification”. Review of the International Statistical Institute 39, 148–169; errata: 40, 221(1972).

    Article  Google Scholar 

  • Gart, J. J., and D. G. Thomas (1972), “Numerical results on approximate confidence limits for the odds ratio”. Journal of the Royal Statistical Society, Series B 34, 441–447.

    MathSciNet  Google Scholar 

  • Gart, J. J., and D. G. Thomas (1982), “The performance of three approximate confidence limit methods for the odds ratio”. American Journal of Epidemiology 115, 453–470.

    Google Scholar 

  • Goodman, L. A. (1969), “On partitioning ×2 nd detecting partial association in three-way contingency tables”. Journal of the Royal Statistical Society, Series B 31, 486–498.

    Google Scholar 

  • Green, S. B., and D. P. Byar (1978), “The effect of stratified randomization on size and power of statistical tests in clinical trials”. Journal of Chronic Diseases 31, 445–454.

    Article  Google Scholar 

  • Greenland, S. (1984), “A counter example to the test-based principle of setting confidence limits”. American Journal of Epidemiology 120, 4–7.

    Google Scholar 

  • Guilbaud, O. (1983), “On the large-sample distribution of the Mantel-Haenszel odds-ratio estimator”. Biometrics 39, 523–525.

    Article  Google Scholar 

  • Haldane, J. B. S. (1955), “The estimation and significance of the logarithm of a ratio of frequencies”. Annals of Human Genetics 20, 309–311.

    Article  Google Scholar 

  • Halperin, M. (1977), “Letter to the editor re: Miettinen (1976)”. American Journal of Epidemiology 105, 496–498.

    Google Scholar 

  • Hannan, J., and W. Harkness (1963), “Normal approximation to the distribution of two independent binomials, conditional on fixed sum”. Annals of Mathematical Statistics 34, 1593–1595.

    Article  MathSciNet  MATH  Google Scholar 

  • Harkness, W. L. (1965), “Properties of the extended hypergeometric distribution”. Annals of Mathematical Statistics 36, 938–945.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Hauck, W. W. (1984), “A comparative study of conditional maximum likelihood estimation of a common odds ratio”. Biometrics 40, 1117–1123.

    Article  Google Scholar 

  • Hauck, W. W., S. Anderson, and F. J. Leahy (1982), “Finite-sample properties of some old and some new estimators of a common odds ratio from multiple 2 × 2 tables”. Journal of the American Statistical Association 77, 145–152.

    Article  MATH  Google Scholar 

  • Hauck, W. W., F. J. Leahy, and S. Anderson (1980), “Finite-sample properties of estimators of a common odds ratio from multiple 2 × 2 tables”. Presented at the August, 1980 Joint Statistical Meetings in Houston, Texas.

    Google Scholar 

  • Hauck, W. W., and C.-B. Wallemark (1983), “A comparison of confidence interval methods for a common odds ratio”. Presented at the August, 1983 Joint Statistical Meetings in Toronto, Ontario.

    Google Scholar 

  • Hitchcock, S. E. (1962), “A note on the estimation of the parameters of the logistic function, using the minimum logit ×2 method”. Biometrika 49, 250–252.

    MATH  Google Scholar 

  • Jewell, N. P. (1984), “Small-sample bias of point estimators of the odds ratio from matched sets”. Biometrics 40, 421–435.

    Article  Google Scholar 

  • Liang, K. Y. (1984), “The comparison of four estimators of a common odds ratio in 2 × 2 tables”. Johns Hopkins University, Department of Biostatistics, paper No. 544.

    Google Scholar 

  • Liang, K. Y. (1985), “Odds ratio reference with dependent data”. Biometrika, 72, 678–682.

    Article  Google Scholar 

  • Lindsay, B. G. (1983), “Efficiency of the conditional score in a mixture setting”. Annals of Statistics 11, 486–497.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Mantel, N. (1977), “Tests and limits for the common odds ratio of several 2 × 2 contingency tables: methods in analogy with the Mantel-Haenszel procedure”. Journal of Statistical Planning and Inference 1, 179–189.

    Article  MATH  Google Scholar 

  • Mantel, N., and W. M. Haenszel (1959), “Statistical aspects of the analysis of data from retrospective studies of disease”. Journal of the National Cancer Institute 22, 719–748.

    Google Scholar 

  • McKinlay, S. M. (1975), “The effects of bias on estimators of relative risk for pair-matched and stratified samples”. Journal of the American Statistical Association 70, 859–864.

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Miettinen, O. S. (1970), “Estimation of relative risk from individually matched series”. Biometrics 26, 75–86.

    Article  Google Scholar 

  • Miettinen, O. (1976), “Estimability and estimation in case-referent studies”. American Journal of Epidemiology 103, 226–235.

    Google Scholar 

  • Miettinen, O. (1977), “Reply to Halperin (1977)”. American Journal of Epidemiology 105, 498–502.

    Google Scholar 

  • Pierce, D. A. (1976), “A random effects model for matched pairs of binomial data”. Technical Report No. 55, Department of Statistics, Oregon State University, Corvallis, Oregon.

    Google Scholar 

  • Pierce, D. A., and B. R. Sands (1975), “Extra-Bernoulli variation in binary data”. Technical Report No. 46, Department of Statistics, Oregon State University, Corvallis, Oregon.

    Google Scholar 

  • Pike, M. C., A. P. Hill, and P. G. Smith (1980), “Bias and efficiency in logistic analyses of stratified case-control studies”. International Journal of Epidemiology 9, 89–95.

    Article  Google Scholar 

  • Robins, J., N. Breslow, and S. Greenland (1986), “Estimators of the Mantel-Haenszel variance consistent in both sparse-data and large-strata limiting models”. Biometrics, in press.

    Google Scholar 

  • Smith, P. G., M. C. Pike, A. P. Hill, N. E. Breslow, and N. E. Day (1981), “Multivariate conditional logistic analysis of stratum-matched case-control studies (Algorithm AS 162)”. Applied Statistics 30, 190–197.

    Article  MATH  Google Scholar 

  • Stiratelli, R., N. Laird, and J. H. Ware (1984), “Random-effects models for serial observations with binary response”. Biometrics 40, 961–971.

    Article  Google Scholar 

  • Tarone, R. E., J. J. Gart, and W. W. Hauck (1983), “On the asymptotic inefficiency of certain noniterative estimators of a common relative risk or odds ratio”. Biometrika 70, 519–522.

    Article  MathSciNet  Google Scholar 

  • Thomas, D. G. (1975), “Exact and asymptotic methods for the combination of 2 × 2 tables”. Computers and Biomedical Research 8, 423–446.

    Article  Google Scholar 

  • Ury, H. K. (1982), “Hauck’s approximate large-sample variance of the Mantel-Haenszel estimator”. Biometrics 38, 1094–1095.

    Google Scholar 

  • Wells, G. A. (1982), “Contributions to the estimation of the logit, log odds and common odds ratio”. Ph.D. thesis, The University of Western Ontario.

    Google Scholar 

  • Woolf, B. (1955), “On estimating the relation between blood group and disease”. Annals of Human Genetics 19, 251–253.

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Epidemiology and International Health, University of California, HSW 1699, San Francisco, California, 94143, USA

    Walter W. Hauck

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  1. Walter W. Hauck
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Editors and Affiliations

  1. Department of Statistical and Actuarial Sciences, The University of Western Ontario, USA

    Ian B. MacNeill , Gary J. Umphrey  & V. Krishna Jandhyala ,  & 

  2. Department of Epidemiology and Biostatistics, The University of Western Ontario, USA

    Allan Donner

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© 1987 D. Reidel Publishing Company, Dordrecht, Holland

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Hauck, W.W. (1987). Estimation of a Common Odds Ratio. In: MacNeill, I.B., Umphrey, G.J., Donner, A., Jandhyala, V.K. (eds) Biostatistics. The University of Western Ontario Series in Philosophy of Science, vol 38. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-4794-8_7

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  • DOI: https://doi.org/10.1007/978-94-009-4794-8_7

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-8626-4

  • Online ISBN: 978-94-009-4794-8

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