A Data-Based Adjustment for Fisher Exact Test

Fisher exact test is one of most popularly used methods in modern data analyses. However, it is conservative because of discreteness. The mid-p method may reduce the conservativeness but it is defined by the factor 1/2, an extra term beyond data. This paper considers an adjustment defined by a data-based factor. The adjusted test is compared with other ten tests. Special attention is given to the comparison between the data-based factor and the factor 1/2. The standardized version of the adjusted test is asymptotically standard normal. The adjustment reduces the conservativeness, as evidenced by increasing test size and power and decreasing p-values. The adjusted test holds such properties as the significance level under control of nominal α, the same modification in the left- and right-sided p-values, and the proportional reduction from Fisher test, which the mid-p method lacks. The mid-p method is more powerful than the adjusted test but the increment of power comes from the factor 1/2 and is not controlled by α. The unconditional tests are also more powerful but the power comes partly from the unobserved samples. The proper choice of an adjustment is based largely upon a consideration of both the power of test and the origin of power so that the adjusted test is an option in data analyses. It is easy to implement for 2 × 2 and r × c contingency tables. Two real examples are given for analyzing 2 × 2 tables and another example for r × c tables.