Approximately Exact Inference for the Common Odds Ratio in Several 2 × 2 Tables

Journal of the American Statistical Association

Volumn 93, Issue 444, 1998, Pages 1294-1307

  Strawderman, Robert L ^a   Wells, Martin T ^b  

^a UNIVERSITY OF MICHIGAN   (United States)

^b Department of Population Medicine and Diagnostic Sciences   (United States)

Author keywords

Conditional inference; Generalized Laplace approximation; Jacobi polynomials; Lugannani Rice approximation; Mantel Haenszel test; Power analysis; P lya frequency sequence; Saddlepoint approximation

Indexed keywords

EID: 0042416886     PISSN: 01621459     EISSN: 1537274X     Source Type: Journal    
DOI: 10.1080/01621459.1998.10473790     Document Type: Article

References (59)

1. Agresti, A. (1992a), “A Survey of Exact Inference for Contingency Tables,” Statistical Science, 7, 131-153.
2. Agresti, A. (1992b), Discussion of “Practical Higher Order Asymptotics for Multiparameter Exponential Families,” Journal of the Royal Statistical Society, Ser. B, 54, 701-737.
3. Agresti, A., Mehta, C. R., and Patel, N. R. (1990), “Exact Inference for Contingency Tables With Ordered Categories,” Journal of the American Statistical Association, 85, 453-458.
4. Aissen, M., Schoenberg, I. J., and Whitney, A. M. (1952), “On the Generating Function of Totally Positive Sequences I,” Journal d'Analyse Mathematique, 2, 93-103.
5. Baglivo, I., Pagano, M., and Spino, C. (1996), “Permutation Distributions via Generating Functions With Applications to Sensitivity Analysis of Discrete Data,” Journal of the American Statistical Association, 91, 1037-1046.
6. Bak, I., and Newman, D. J. (1982), Complex Analysis, New York: SpringerVerlag.
7. Bini, D., and Pan, V. (1991), “Parallel Complexity of Tridiagonal Symmetric Eigenvalue Problems,” in Proceedings of the 2nd Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 384-393.
8. Booth, J. G., and Butler, R. W. (1990), “Randomization Distributions and Saddlepoint Approximations in Generalized Linear Models,” Biometrika, 77, 787-796.
9. Brenti, F. (1989), “Unimodal, Log-Concave, and Polya Frequency Sequences in Combinatorics,” Memoirs of the American Mathematical Society, 413.
10. Breslow, N. (1981), “Odds Ratio Estimators When the Data Are Sparse,” Biometrika, 68, 73-84.
11. Breslow, N., and Liang, K.-Y. (1982), 'The Variance of the Mantel-Haenszel Estimator,” Biometrics, 38, 943-952.
12. Butler, R. W., Huzurbazar, S., and Booth, I. G. (1992), “Saddlepoint Approximations for the Bartlett-Nanda-Pillai Trace Statistics in Multivariate Analysis,” Biometrika, 79, 705-716.
13. Casella, G., and Berger, R. (1990), Statistical Inference, Pacific Grove, CA: Wadsworth & Brooks/Cole.
14. Cox, D. R., and Snell, E. J. (1989), Analysis of Binary Data (2nd ed.), London: Chapman and Hall.
15. Cytel Software Corporation (1992), StatXact: A Statistical Package for Exact Nonparametric Inference, Ver. 2.01, Cambridge, MA: Author.
16. Daniels, H. E. (1987), “Tail Probability Approximations,” International Statistical Review, 55, 37-48.
17. Davis, L. J. (1985), “Generalization of the Mantel-Haenszel Estimator to Nonconstant Odds Ratios,” Biometrics, 41, 487-495.
18. Davison, A. (1988), “Approximate Conditional Inference in Generalized Linear Models,” Journal of the Royal Statistical Society, Ser. B, 50, 445-461.
19. DiCiccio, T. I., and Martin, M. (1991), “Approximations of Marginal Tail Probabilities for a Class of Smooth Functions With Applications to Bayesian and Conditional Inference,” Biometrika, 78, 891-902.
20. Everitt, B. S. (1992), The Analysis of Contingency Tables (2nd ed.), London: Chapman and Hall.
21. Gastwirth, J. L. (1988), Statistical Reasoning in Law and Public Policy, Vol. 1, New York: Academic Press.
22. Goedecker, S. (1994), “Remark on Algorithms to Find Roots of Polynomials,” SIAM Journal on Scientific Computing, 15, 1059-1063.
23. Harper, L. H. (1966), “Stirling Behavior Is Asymptotically Normal,” Annals of Mathematical Statistics, 38, 410-414.
24. Hauck, W. (1982), “A Comparative Study of the Conditional Maximum Likelihood Estimation of a Common Odds Ratio,” Biometrics, 40, 1117-1123.
25. Hirji, K. F., Mehta, C. R., and Patel, N. R. (1987), “Computing Distributions for Exact Logistic Regression,” Journal of the American Statistical Association, 82, 1110-1117.
26. Hirji, K. F., Tang, M., Vollset, S. E., and Elashoff, R. M. (1994), “Efficient Power Computation for Exact and Mid-p Tests for the Common Odds Ratio in Several 2x2 Tables,” Statistics in Medicine, 13, 1539-1549.
27. Hwang, J. T. G., and Yang, M.-C. (1998), “Evaluating the p-Values for Testing the Independence in 2 x 2 Contingency Tables Using the Estimated Truth Approach-One Way To Resolve the Controversy Related to Fisher's Exact Test,” unpublished manuscript submitted to Statistica Sinica.
28. Jensen, J. L. (1995), Saddlepoint Approximations, Oxford, U.K.: Clarendon Press.
29. Kim, D., and Agresti, A. (1995), “Improved Exact Inference About Conditional Association in Three-Way Contingency Tables,” Journal of the American Statistical Association, 90, 632-639.
30. Kolassa, I. E. (1994), Series Approximation Methods in Statistics, New York: Springer-Verlag.
31. Kolassa, I. E., and Tanner, M. A. (1994), “Approximate Conditional Inference via the Gibbs Sampler,” Journal of the American Statistical Association, 81, 278-288.
32. Kou, S., and Ying, Z. (1996a), “Asymptotics for a 2 x 2 Table With Fixed Margins,” Statistica Sinica, 6, 809-830.
33. Kou, S., and Ying, Z. (1996b), “Analysis of Sequence of 2 x 2 Tables,” unpublished technical report.
34. Lancaster, H. O. (1961), “Significance Tests in Discrete Distributions,” Journal of the American Statistical Association, 56, 223-234.
35. Lehmann, E. L. (1986), Testing Statistical Hypotheses (2nd ed.), New York: Wiley.
36. Levy, P. (1937), “Sur Une Propriété de la loi de Poisson Relative aux Petites Probabilités,” Société Mathématiques de France, Comptes Randus des Scéances de l'Annee 1936, p. 29.
37. Lugannani, R., and Rice, S. (1980), “Saddlepoint Approximation for the Distribution of the Sum of Independent Random Variables,” Advances in Applied Probability, 12, 475-490.
38. Mantel, N., and Haenszel, W. M. (1959), “Statistical Aspects of the Analysis of Data From Retrospective Studies of Disease,” Journal of the National Cancer Institute, 22, 719-748.
39. McCullagh, P., and Nelder, J. A. (1989), Generalized Linear Models (2nd ed.), London: Chapman and Hall.
40. Mehta, C. R., and Patel, N. R. (1983), “A Network Algorithm for Performing Fisher's Exact Test in R x C Contingency Tables,” Journal of the American Statistical Association, 78, 427-434.
41. Mehta, C. R., Patel, N. R., and Gray, R. (1985), “Computing an Exact Confidence Interval for the Common Odds Ratio in Several 2x2 Contingency Tables,” Journal of the American Statistical Association, 80, 969-973.
42. Mehta, C. R., Patel, N. R., and Senchaudhuri, P. (1988), “Importance Sampling for Estimating Exact Probabilities in Permutational Inference,” Journal of the American Statistical Association, 83, 999-1005.
43. Mehta, C. R., Patel, N. R., and Senchaudhuri, P. (1992), “Exact Stratified Linear Rank Tests for Ordered Categorical and Binary Data,” Journal of Computational and Graphical Statistics, 1, 21-40.
44. Mehta, C. R., and Walsh, S. J. (1992), “Comparison of Exact, Mid-p, and Mantel-Haenszel Confidence Intervals for the Common Odds Ratio Across Several 2x2 Tables,” The American Statistician, 46, 146-151.
45. Munoz, A., and Rosner, B. (1984), “Power and Sample Size for a Collection of 2 x 2 Tables,” Biometrics, 40, 995-1004.
46. Olver, F. W. J. (1974), Asymptotics and Special Functions, New York: Academic Press.
47. Pan, V. (1997), “Solving a Polynomial Equation: Some History and Recent Progress,” SIAM Review, 39, 187-220.
48. Parlett, B. (1980), The Symmetric Eigenvalue Problem, Englewood Cliffs, NJ: Prentice Hall.
49. Pierce, D., and Peters, D. (1992), “Practical Higher Order Asymptotics for Multiparameter Exponential Families,” Journal of the Royal Statistical Society, Ser. B, 54, 701-737.
50. Pitman, J. (1997), “Probabilistic Bounds on the Coefficients of Polynomials With Only Real Zeros,” Journal of Combinatorial Theory, Ser. A, 77, 279-303.
51. Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P. (1992), Numerical Recipes (2nd ed.), Cambridge, U.K.: Cambridge University Press.
52. Robins, J., Breslow, N., and Greenland, S. (1986), “Estimators of the Mantel-Haenszel Variance Consistent in Both Sparse Data and Large-Strata Limiting Models,” Biometrics, 42, 311-325.
53. Schoenberg, I. J. (1955), “On The Zeros of the Generating Functions of Multiply Positive Sequences and Functions,” Annals of Mathematics, 62, 447-471.
54. Strawderman, R. L., Casella, G., and Wells, M. T. (1996), “Practical Small-Sample Asymptotics for Regression Problems,” Journal of the American Statistical Association, 91, 643-655.
55. Szego, G. (1975), Orthogonal Polynomials, Providence, RI: American Mathematical Society.
56. Temme, N. (1982), 'The Uniform Asymptotic Expansion of a Class of Integrals Related to Cumulative Distribution Functions,” SIAM Journal of Mathematical Analysis, 13, 239-253.
57. van Doom, E. A. (1987), “Representations and Bounds for Zeros of Orthogonal Polynomials and Eigenvalues of Sign-Symmetric Tridiagonal Matrices,” Journal of Approximation Theory, 51, 254-266.
58. Vollset, S. E., Hirji, K. F., and Elashoff, R. M. (1991), “Fast Computation of Exact Confidence Limits for the Common Odds Ratio in a Series of 2 x 2 Tables,” Journal of the American Statistical Association, 86, 404-409.
59. Wang, S. (1993), “Saddlepoint Expansions in Finite Population Problems,” Biometrika, 80, 583-590.