# 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 .