Stratified Fisher’s exact test and its sample size calculation. (English) Zbl 1336.62059

Summary: Chi-squared test has been a popular approach to the analysis of a \(2 \times 2\) table when the sample sizes for the four cells are large. When the large sample assumption does not hold, however, we need an exact testing method such as Fisher’s test. When the study population is heterogeneous, we often partition the subjects into multiple strata, so that each stratum consists of homogeneous subjects and hence the stratified analysis has an improved testing power. While Mantel-Haenszel test has been widely used as an extension of the chi-squared test to test on stratified \(2 \times 2\) tables with a large-sample approximation, we have been lacking an extension of Fisher’s test for stratified exact testing. In this paper, we discuss an exact testing method for stratified \(2 \times 2\) tables that is simplified to the standard Fisher’s test in single \(2 \times 2\) table cases, and propose its sample size calculation method that can be useful for designing a study with rare cell frequencies.
Editorial remark: See also the comment of A. Martín-Andrés and I. Herranz-Tejedor [Biom. J. 57, No. 5, 930 (2015; Zbl 1336.62062)].


62F03 Parametric hypothesis testing


Zbl 1336.62062
Full Text: DOI Link


[1] Breslow, N. E. and Day, N. E. (1980). The Analysis of Case‐Control Studies. No. 32, IARC Scientific Publications, Lyon, France.
[2] Cochran, W. C. (1954). Some methods of strengthening the common χ^2 tests. Biometrics10, 417-451. · Zbl 0059.12803
[3] Crans, G. G. and Schuster, J. J. (2008). How conservative is Fisher’s exact test? A quantitave evaluation of the two‐sample comparative binomiial trial. Statistics in Medicine27, 3598-3611.
[4] Fisher, R. A. (1935). The logic of inductive inference (with discussion). Journal of Royal Statistical Society98, 39-82. · Zbl 0011.03205
[5] Gart, J. J. (1985). Approximate tests and interval estimation of the common relative risk in the combination of 2 × 2 tables. Biometrika72, 673-677.
[6] Jung, S. H., Chow, S. C. and Chi, E. M. (2007). A note on sample size calculation based on propensity analysis in nonrandomized trials. Journal of Biopharmaceutical Statistics17, 35-41.
[7] Li, S. H., Simon, R. M. and Gart, J. J. (1979). Small sample properties of the Mantel-Haenszel test. Biometrika66, 181-183.
[8] Mantel, N. and Haenszel, W. (1959). Statistical aspects of the analysis of data from retrospective studies of disease. Journal of the National Cancer Institute22, 719-748.
[9] Nam, J. M. (1992). Sample size determination for case‐control studies and the comparison of stratified and unstratified analyses. Biometrics48, 389-395.
[10] Nam, J. M. (1998). Power and sample size for stratified prospective studies using the score method for testing relative risk. Biometrics54, 331-336. · Zbl 1058.62566
[11] Westfall, P. H., Zaykin, D. V. and Young, S. S. (2002). Multiple tests for genetic effects in association studies. In: StephenLooney (ed.) (Ed.), Methods in Molecular Biology, , Biostatistical Methods, Humana Press, Toloway, NJ, pp. 143-168.
[12] Woolson, R. F., Bean, J. A. and Rojas, P. B. (1986). Sample size for case‐control studies using Cochran’s statistic. Biometrics42, 927-932. · Zbl 0657.62124
[13] Zelen, M. (1971). The analyses of several 2x2 contingency tables. Biometrika58, 129-137. · Zbl 0218.62061
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