Issues in biomedical statistics: analysing 2 x 2 tables of frequencies

How best to analyse statistically experimental results that are set out as a 2 x 2 table of frequencies has been debated by statisticians for more than 50 years. The main issue is what framework of statistical inference should be adopted. The design of most biomedical experiments that result in 2 x 2 tables of independent observations is compatible with the randomization model of inference and with the Fisher exact test. It is rare that the Neyman-Pearson population model is applicable and that a case can be made for using the Pearson chi 2 test, or others that refer a test statistic to the chi-squared distribution. Even then, the adjustments for the mismatch between the test statistic and the chi-squared distribution so as to control the risk of Type I error are complex that the Fisher test is probably a safer option (or Yates' correction to the Pearson test if there is no access to a computer). When the 2 x 2 table results from two sets of measurements having been made on the same group, the population model of inference is inapplicable and the exact form of the McNemar test should be used. Confidence intervals for differences in proportions, the likelihood ratio, or the odds ratio, refer to randomly sampled populations and are not compatible with the randomization model of inference.