The POWER Procedure
Analyses in the TWOSAMPLEFREQ Statement
Overview of the 
Table
Notation:
| Outcome | ||||
| Failure | Success | |||
| Group | 1 |  |
 |
 |
| 2 |  |
 |
 |
|
 |
m | N | ||
The hypotheses are
where 
is constrained to be 0 for the likelihood ratio and Fisher’s exact tests. If 
in an upper one-sided test or 
in a lower one-sided test, then the test is a noninferiority test. If in an upper one-sided test or in a lower one-sided test, then the test is a superiority test. Although is unconstrained for the Pearson chi-square test, 
is not recommended for that test. The Farrington-Manning score test is a better choice when .
Internal calculations are performed in terms of 
, 
, and . An input set consisting of OR, , and 
is transformed as follows:
An input set consisting of RR, , and 
is transformed as follows:
The transformation of either or to is not unique. The chosen parameterization fixes the null value 
at the input value of . Some values of or might lead to invalid values of (
or 
), in which case an "Invalid input" error occurs.
Farrington-Manning Score Test for Proportion Difference (TEST=FM)
The Farrington-Manning score test for proportion difference is based on equations (1), (2), and (12) in Farrington and Manning (1990). The test statistic, which is assumed to have a null distribution of 
under 
, is
where 
and 
are the maximum likelihood estimates of the proportions under the restriction 
.
Sample size for the one-sided cases is given by equations (4) and (12) in Farrington and Manning (1990). One-sided power is computed by inverting the sample size formula. Power for the two-sided case is computed by adding the lower-sided and upper-sided powers, each evaluated at 
. Sample size for the two-sided case is obtained by numerically inverting the power formula,
where
For the one-sided cases, a closed-form inversion of the power equation yields an approximate total sample size of
For the two-sided case, the solution for N is obtained by numerically inverting the power equation.
Farrington-Manning Score Test for Relative Risk (TEST=FM_RR)
The Farrington-Manning score test is based on equations (5), (6), and (13) in Farrington and Manning (1990). The test statistic, which is assumed to have a null distribution of under , is
where and are the maximum likelihood estimates of the proportions under the restriction 
.
Sample size for the one-sided cases is given by equations (8) and (13) in Farrington and Manning (1990). One-sided power is computed by inverting the sample size formula. Power for the two-sided case is computed by adding the lower-sided and upper-sided powers, each evaluated at . Sample size for the two-sided case is obtained by numerically inverting the power formula,
where
For the one-sided cases, a closed-form inversion of the power equation yields an approximate total sample size of
For the two-sided case, the solution for N is obtained by numerically inverting the power equation.
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 evaluated at . 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, but it is not recommended. If you are using a nondefault null value, then the Farrington-Manning score test is a better choice.
For the one-sided cases, a closed-form inversion of the power equation yields an approximate total sample size
For the two-sided case, the solution for N 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 N 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 m. 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:
The approximation is valid only for 
.