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Appendix 5: Significance Levels for
Fisher’s Exact Test
The procedure described in this appendix is used to calculate the exact one-tailed and two-tailed significance levels of Fisher’s exact test for a 2 × 2 table under the assumption of independence of rows and columns and conditional on the marginal totals. All cell counts are rounded to the nearest integers.
Background
Consider the following observed 2 × 2 table:
n 1 n 2 n (^) 1 + n 2 n 3 n 4 n (^) 3 + n 4 n 1 (^) + n 3 n (^) 2 + n 4 N Conditional on the observed marginal totals, the values of the four cell counts can be expressed as the observed count of the first cell n 1 only. Under the hypothesis of independence, the count of the first cell N 1 follows a hypergeometric distribution with the probability of N (^) 1 = n 1 given by
Prob N n
n n n n n n n n (^1 1) N n n n n 1 2 3 4 1 3 2 4 1 2 3 4
1 6 1!^ 6 1!^ 6 1!^6!
where N 1 ranges from max 10 , n 1 − n 46 to min 1 n 1 + n 2 , n 1 + n 36 and
N = n 1 (^) + n (^) 2 + n (^) 3 + n 4. The exact one-tailed significance level p 1 is defined as
p N n n E N (^1) N n n E N
1 1 1 1 1 1 1 1
K
'K
Prob if Prob if
1 This algorithm applies to SPSS 6.1.2 and later releases.
2 Appendix 5
where E N 1 6 1 1 (^) = n (^) 1 + n (^) 2 61 n (^) 1 + n 3 (^) 6 / N. The exact two-tailed significance level p 2 is defined as the sum of the one- tailed significance level p 1 and the probabilities of all points in the other side of the sample space of N 1 which are not greater than the probability of N (^) 1 = n 1.
Computations
To begin the computation of the two significance levels p 1 and p 2 , the counts in the observed 2 × 2 table are rearranged. Then the exact one-tailed and two-tailed significance levels are computed using the CDF.HYPER cumulative distribution function.
Table Rearrangement
The following steps are used to rearrange the table:
- Check whether n 1 (^) > E N 1 6 1 , which can be done by checking whether n n 1 (^) 4 > n n 2 3. If so, rearrange the table so that the first cell contains the minimum of n 2 and n 3 , maintaining the row and column totals; otherwise, rearrange the table so that the first cell contains the minimum of n 1 and n 4 , again maintaining the row and column totals.
- Without loss of generality, we assume that the count of the first cell is n 1 after the above rearrangement. Calculate the first row total, the first column total, and the overall total, and name them SAMPLE , HITS , and TOTAL , respectively.
One-Tailed Significance Level
The following steps are used to calculate the one-tailed significance level:
- If TOTAL = 0 , set the one-tailed significance level p 1 equal to 1; otherwise, obtain p 1 by using the CDF.HYPER cumulative distribution function with arguments n 1 , SAMPLE , HITS , and TOTAL.
- Also calculate the probability of the first cell count equal to n 1 by finding the difference between p 1 and the value obtained from CDF.HYPER with n 1 − 1 , SAMPLE , HITS , and TOTAL as its arguments, provided that n 1 > 0. Call this probability PEXACT.
- If n 1 = 0 , set PEXACT = p 1. PEXACT will be used in the next step to find the points for which the probabilities are not greater than PEXACT.
