Notation:
Outcome 

Failure 
Success 

Group 
1 



2 







The hypotheses are
where is constrained to be for all but the unconditional Pearson chisquare test.
Internal calculations are performed in terms of , , and . An input set consisting of , , and is transformed as follows:
An input set consisting of , , and is transformed as follows:
Note that the transformation of either or to is not unique. The chosen parameterization fixes the null value at the input value of .
The usual Pearson chisquare test is unconditional. The test statistic
is assumed to have a null distribution of .
Sample size for the onesided cases is given by equation (4) in Fleiss, Tytun, and Ury (1980). Onesided power is computed as suggested by Diegert and Diegert (1981) by inverting the sample size formula. Power for the twosided case is computed by adding the lowersided and uppersided powers each with , and sample size for the twosided case is obtained by numerically inverting the power formula. A custom null value for the proportion difference is also supported.
For the onesided cases, a closedform inversion of the power equation yield an approximate total sample size
For the twosided case, the solution for is obtained by numerically inverting the power equation.
The usual likelihood ratio chisquare test is unconditional. The test statistic
is assumed to have a null distribution of and an alternative distribution of , where
The approximate power is
For the onesided cases, a closedform inversion of the power equation yield an approximate total sample size
For the twosided case, the solution for is obtained by numerically inverting the power equation.
Fisher’s exact test is conditional on the observed total number of successes . Power and sample size computations are based on a test with similar power properties, the continuityadjusted arcsine test. The test statistic
is assumed to have a null distribution of and an alternative distribution of , where
The approximate power for the onesided balanced case is given by Walters (1979) and is easily extended to the unbalanced and twosided cases: