## The $$2\times2$$ table: A discussion from a Bayesian viewpoint.(English)Zbl 1059.62526

Summary: The $$2\times2$$ table is used as a vehicle for discussing different approaches to statistical inference. Several of these approaches (both classical and Bayesian) are compared, and difficulties with them are highlighted. More frequent use of one-sided tests is advocated. Given independent samples from two binomial distributions, and taking independent Jeffreys priors, we note that the posterior probability that the proportion of successes in the first population is larger than in the second can be estimated from the standard (uncorrected) chi-square significance level. An exact formula for this probability is derived. However, we argue that usually it will be more appropriate to use dependent priors, and we suggest a particular ”standard prior” for the $$2\times2$$ table. For small numbers of observations this is more conservative than Fisher’s exact test, but it is less conservative for larger sample sizes. Several examples are given.

### MSC:

 62F15 Bayesian inference 62H17 Contingency tables
Full Text:

### References:

 [1] Abramowitz, M. and Stegun, I., eds. (1970). Handbook of Mathematical Functions. National Bureau of Standards, U.S. Government Printing Office, Washington, DC. · Zbl 0171.38503 [2] Altham, P. M. E. (1969). Exact Bayesian analysis of a 2 \times 2 contingency table, and Fisher’s ”exact” significance test. J. Roy. Statist. Soc. Ser. B 31 261-269. JSTOR: [3] Antelman, G. R. (1972). Interrelated Bernoulli processes. J. Amer. Statist. Assoc. 67 831-841. · Zbl 0271.62023 [4] Barnard, G. A. (1947). Significance tests for 2 \times 2 tables. Biometrika 34 123-138. Bartlett, R. H., Roloff, D. W., Cornell, R. G., Andrews, A. F., JSTOR: · Zbl 0029.15603 [5] Dillon, P. W. and Zwischenberger, J. B. (1985). Extracorporeal circulation in neonatal respiratory failure: a prospective randomised study. Pediatrics 76 479-487. [6] Begg, C. B. (1990). On inferences from Wei’s biased coin design for clinical trials. Biometrika 77 467-484. JSTOR: · Zbl 0716.62108 [7] Berger, J. O., Boukai, B. and Wang, Y. (1997). Unified frequentist and Bayesian testing of a precise hy pothesis (with discussion). Statist. Sci. 12 133-160. · Zbl 0955.62527 [8] Berger, J. O. and Delampady, M. (1987). Testing precise hy potheses. Statist. Sci. 2 317-352. · Zbl 0955.62545 [9] Berkson, J. (1978). In dispraise of the exact test. J. Statist. Plann. Inference 2 27-42. · Zbl 0374.62029 [10] Church, A. (1940). On the concept of a random sequence. Bull. Amer. Math. Soc. 46 130-135. · Zbl 0022.36904 [11] Conover, W. J. (1974). Some reasons for not using the Yates continuity correction on 2 \times 2 contingency tables (with discussion). J. Amer. Statist. Assoc. 69 374-382. Cornell, R. G., Landenberger, B. D. and Bartlett, R. H. [12] . Randomized play the winner clinical trials. Comm. Statist. Theory Methods 15 159-178. · Zbl 0607.62095 [13] Cornfield, J. (1956). A statistical problem arising from retrospective studies. Proc. Third Berkeley Sy mp. Math. Statist. Probab. 135-148. Univ. California Press, Berkeley. · Zbl 0070.14707 [14] Cox, D. R. and Hinkley, D. V. (1974). Theoretical Statistics. Chapman and Hall, London. · Zbl 0334.62003 [15] D’Agostino, R. B., Chase, W. and Belanger, A. (1988). The appropriateness of some common procedures for testing the equality of two independent binomial proportions. Amer. Statist. 42 198-202. [16] Dawid, A. P. (1984). Statistical theory: the prequential approach (with discussion). J. Roy. Statist. Soc. Ser. A 147 278-292. JSTOR: · Zbl 0557.62080 [17] Edwards, A. W. F. (1972). Likelihood. Cambridge Univ. Press. · Zbl 0231.62005 [18] Fisher, R. A. (1935). The Design of Experiments. Oliver and Boy d, Edinburgh. · Zbl 0011.03205 [19] Fisher, R. A. (1945). A new test for 2\times 2 tables. Nature 156 388. [20] Franck, W. E. (1986). P-values for discrete test statistics. Biometrical J. 28 403-406. [21] Fraser, D. A. S., Monette, G. and Ng, K.-W. (1984). Marginalization, likelihood and structural models. In Multivariate Analy sis VI (P. R. Krishnaiah, ed.) 209-217. North-Holland, Amsterdam. · Zbl 0594.62001 [22] Goldstein, M. and Howard, J. V. (1991). A likelihood paradox. J. Roy. Statist. Soc. Ser. B 53 619-628. JSTOR: · Zbl 0800.62029 [23] Grizzle, J. E. (1967). Continuity correction in the 2-test for 2 \times 2 tables. Amer. Statist. 21 28-32. [24] Haber, M. (1986). A modified exact test for 2 \times 2 contingency tables. Biometrical J. 28 455-463. · Zbl 0607.62061 [25] Jeffrey s, H. (1961). Theory of Probability, 3rd ed. Oxford Univ. Press. · Zbl 0116.34904 [26] Kass, R. E. and Raftery, A. E. (1995). Bay es factors. J. Amer. Statist. Assoc. 90 773-795. · Zbl 0846.62028 [27] Lancaster, H. O. (1961). Significance tests in discrete distributions. J. Amer. Statist. Assoc. 56 233-234. JSTOR: · Zbl 0104.13201 [28] Lane, D. A. and Sudderth, W. D. (1983). Coherent and continuous inference. Ann. Statist. 11 114-120. · Zbl 0563.62003 [29] Lehmann, E. L. (1986). Testing Statistical Hy potheses, 2nd ed. Wiley, New York. [30] Little, R. J. A. (1989). Testing the equality of two independent binomial proportions. Amer. Statist. 43 283-288. [31] Pearson, E. S. (1947). The choice of statistical tests illustrated on the interpretation of data classed in a 2 \times 2 table. Biometrika 34 139-167. JSTOR: · Zbl 0029.27406 [32] Pearson, K. (1900). On the criterion that a given sy stem of deviations from the probable in the case of a correlated sy stem of variables is such that it can be reasonably supposed to have arisen from random sampling. Phil. Mag. 5(50) 157-175. · JFM 31.0238.04 [33] Pratt, J. W. (1965). Bayesian interpretation of standard inference statements (with discussion). J. Roy. Statist. Soc. Ser. B 27 169-203. JSTOR: · Zbl 0142.15203 [34] Robbins, H. (1977). A fundamental question of practical statistics. Letter to the editor. Amer. Statist. 31 97. [35] Routledge, R. D. (1992). Resolving the conflict over Fisher’s exact test. Canad. J. Statist. 20 201-209. JSTOR: · Zbl 0766.62031 [36] Stone, M. (1969). The role of significance testing: some data with a message. Biometrika 56 485-493. · Zbl 0183.48105 [37] Suissa, S. and Shuster, J. J. (1985). Exact unconditional sample sizes for the 2 \times 2 binomial trial. J. Roy. Statist. Soc. Ser. A 148 317-327. JSTOR: · Zbl 0585.62101 [38] Tocher, K. D. (1950). Extension of Ney man-Pearson theory of tests to discontinuous variates. Biometrika 37 130-144. JSTOR: · Zbl 0040.07502 [39] Upton, G. J. G. (1992). Fisher’s exact test. J. Roy. Statist. Soc. Ser. A 155 395-402. [40] Wei, L. J. (1988). Exact two-sample permutation tests based on the randomized play-the-winner rule. Biometrika 75 603- 606. JSTOR: · Zbl 0657.62086 [41] Wei, L. J. and Durham, S. (1978). The randomized play-thewinner rule in medical trials. J. Amer. Statist. Assoc. 73 830- 843. · Zbl 0389.62067 [42] Yates, F. (1984). Tests of significance for 2\times 2 contingency tables (with discussion). J. Roy. Statist. Soc. Ser. A 147 426-463. · Zbl 0573.62050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.