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# Fisher Exact

### Input.

Fill the values in the table. Are considered to be integer values, whole positive numbers without decimals.

A windows version of the Fisher procedure is available here.

### Explanation.

The Fisher's Exact test procedure calculates an exact probability value for the relationship between two dichotomous variables, as found in a two by two crosstable. The program calculates the difference between the data observed and the data expected, considering the given marginal and the assumptions of the model of independence. It works in exactly the same way as the Chi-square test for independence; however, the Chi-square gives only an estimate of the true probability value, an estimate which might not be very accurate if the marginal is very uneven or if there is a small value (less than five) in one of the cells. In such cases the Fisher exact test is a better choice than the Chi-square. However, in many cases the Chi-square is preferred because the Fisher exact test is difficult to calculate.

The one-sided probability for the Fisher exact test is calculated by generating all tables that are more extreme than the table given by the user, in one direction. The p-values of these tables are added up, including the p-value of the table itself. The single-sided p-value is the summed probability of all more extreme or similar tables compared with the given table (notation p(Observed>=Expected )). There is also a p-value of the relationship going in the other direction. This is calculated by taking all the tables which are more extreme or the same in the opposite direction (notation: O<=E). Do not take too much notice of this p-value; it is not so important. However, it is mentioned in the textbooks and SISA does not want to deprive you of this information.

Exact p-values tend to be more conservative than most approximate estimates, such as the Chi-square or the t-test. To make the exact p-value more like the approximate result the mid-p value is sometimes used. The mid-p is somewhere in the middle between p-values including the point probability values p(0>=E) and p-values not including the point probability values p(0>E). To determine the mid-p value SISA takes the average of these two values. SISA does not recommend the use of mid-p values.

There are a number of theories about how to present double-sided p-values (Agresti, 1992). Data on the basis of two of these theories are presented. First, the sum of small p-values. For the sum of small p-values all tables are generated which are possible given the margins. All p-values of the same size or smaller than the point probability are added up to form the cumulative p-value. The result is relevant to the notation p(O>=E |O<=E). Statisticians usually recommend this method. Another method of estimating the double-sided p-value is to take twice the single-sided probability. The notation for this method is the same as for the method of small p-values. Simulations show that the p-value for the method of small p-values not including the point probability {p(O>E|O<E)} is often closer to the Chi-square. You can compare the relevant p-values with two Chi-square tests yourself. Lastly, note that the difference between p(O>=E |O<=E) and p(O>E|O<E) and the problem of which p-value to compare with the Chi-square is dependent on the amount of continuity inherent in the table. The Chi-square is based on a continuous distribution while in fact the p-values 'jump' in value between tables. In the case of the Fisher this jumping is dependent on the number of different tables that can be produced given the margins. The number of tables, and therefore the level of continuity, is not dependent on the total sample size but on the number of cases in the smallest marginal.

### Median Test

One particular application of the Fisher exact is as a test for the difference in location between two medians (the median is the point or value above which we can find exactly half of the observations). The null hypothesis is that the medians for two groups are the same, the alternative is that the locations are different. Fill in the number of scores for group one above the combined median (the one for both groups taken together), then do the same for group two. See the example. The Fisher exact probability values are valid for the median test (in this case 0.003047 single-sided or 0.006095 double-sided). Group 1 is statistically significantly different from Group 2. Note: use double-sided testing if there was no prior expectation as to which of the two groups had a higher median.

 example data for median test Group 1 Group 2 Combined No of scores above combined median No of scores below combined median 9 21 24 12 33 33

Learn more about the Fisher-Exact-test from Statistics at Square One. 