### Analyses in the TWOSAMPLEFREQ Statement

#### Overview of the Table

Notation:

 Outcome Failure Success Group 1   2                  The hypotheses are    where is constrained to be for all but the unconditional Pearson chi-square 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 .

#### Pearson Chi-Square Test for Two Proportions (TEST=PCHI)

The usual Pearson chi-square test is unconditional. The test statistic is assumed to have a null distribution of .

Sample size for the one-sided cases is given by equation (4) in Fleiss, Tytun, and Ury (1980). One-sided power is computed as suggested by Diegert and Diegert (1981) by inverting the sample size formula. Power for the two-sided case is computed by adding the lower-sided and upper-sided powers each with , and sample size for the two-sided case is obtained by numerically inverting the power formula. A custom null value for the proportion difference is also supported. For the one-sided cases, a closed-form inversion of the power equation yield an approximate total sample size For the two-sided case, the solution for is obtained by numerically inverting the power equation.

#### Likelihood Ratio Chi-Square Test for Two Proportions (TEST=LRCHI)

The usual likelihood ratio chi-square 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 one-sided cases, a closed-form inversion of the power equation yield an approximate total sample size For the two-sided case, the solution for is obtained by numerically inverting the power equation.

#### Fisher’s Exact Conditional Test for Two Proportions (Test=FISHER)

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 continuity-adjusted arcsine test. The test statistic    is assumed to have a null distribution of and an alternative distribution of , where    The approximate power for the one-sided balanced case is given by Walters (1979) and is easily extended to the unbalanced and two-sided cases: 