The POWER Procedure
Analyses in the ONECORR Statement
Fisher’s z Test for Pearson Correlation (TEST=PEARSON DIST=FISHERZ)
Fisher’s z transformation (Fisher 1921) of the sample correlation 
is defined as
Fisher’s z test assumes the approximate normal distribution 
for z, where
and
where 
is the number of variables partialed out (Anderson 1984, pp. 132–133) and 
is the partial correlation between Y and 
adjusting for the set of zero or more variables 
.
The test statistic
is assumed to have a normal distribution 
, where 
is the null partial correlation and 
and 
are derived from Section 16.33 of Stuart and Ord (1994):
The approximate power is computed as
Because the test is biased, the achieved significance level might differ from the nominal significance level. The actual alpha is computed in the same way as the power, except that the correlation is replaced by the null correlation .
t Test for Pearson Correlation (TEST=PEARSON DIST=T)
The two-sided case is identical to multiple regression with an intercept and 
, which is discussed in the section Analyses in the MULTREG Statement.
Let denote the number of variables partialed out. For the one-sided cases, the test statistic is
which is assumed to have a null distribution of 
.
If the X and Y variables are assumed to have a joint multivariate normal distribution, then the exact power is given by the following formula:
The distribution of (given the underlying true correlation ) is given in Chapter 32 of Johnson, Kotz, and Balakrishnan (1995).
If the X variables are assumed to have fixed values, then the exact power is given by the noncentral t distribution 
, where the noncentrality is
The power is
