test is exact only for the conditional distribution and not the original table where the margin totals may change from experiment to experiment. It is possible to obtain an exact p-value for the 2×2 table when the margins are not held fixed. Barnard's test, for example, allows for random margins. However, some authors (including, later, Barnard himself) have criticized Barnard's test based on this property. They argue that the marginal success total is an (almost) ancillary statistic, containing (almost) no information about the tested property. The act of conditioning on the marginal success rate from a 2×2 table can be shown to ignore some information in the data about the unknown odds ratio. The argument that the marginal totals are (almost) ancillary implies that the appropriate likelihood function for making inferences about this odds ratio should be conditioned on the marginal success rate. Whether this lost information is important for inferential purposes is the essence of the controversy. Alternatives An alternative exact test, Barnard's exact test, has been developed and proponents of it suggest that this method is more powerful, particularly in 2×2 tables. Furthermore, Boschloo's test is an exact test that is uniformly more powerful than Fisher's exact test by construction. Another alternative is to use maximum likelihood estimates to calculate a p-value from the exact binomial or multinomial distributions and reject or fail to reject based on the p-value. For stratified categorical data the Cochran–Mantel–Haenszel test must be used instead of Fisher's test. Choi et al. propose a p-value derived from the likelihood ratio test based on the conditional distribution of the odds ratio given the marginal success rate. This p-value is inferentially consistent with classical tests of normally distributed data as well as with likelihood ratios and support intervals based on this conditional likelihood function. It is also readily computable. See also Bernoulli trial Boschloo's test References External links Calculate Fisher's Exact Test Online Likelihood Ratio Statistics for 2X2 Tables Statistical tests for contingency tables Nonparametric statistics