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| The POWER Procedure |
Analyses in the TWOSAMPLEFREQ Statement
Overview of the 
Table
Notation:
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The hypotheses are
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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:
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An input set consisting of 
, , and 
is transformed as follows:
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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
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is assumed to have a null distribution of 
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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.
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For the one-sided cases, a closed-form inversion of the power equation yield an approximate total sample size
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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
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is assumed to have a null distribution of and an alternative distribution of 
, where
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The approximate power is
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For the one-sided cases, a closed-form inversion of the power equation yield an approximate total sample size
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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 for the METHOD=WALTERS option are based on a test with similar power properties, the continuity-adjusted arcsine test. The test statistic
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is assumed to have a null distribution of and an alternative distribution of , where
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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:
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