For a sample correlation r that uses a sample from a bivariate normal distribution with correlation , the statistic
has a Student’s t distribution with (n-2) degrees of freedom.
With the monotone transformation of the correlation r (Fisher 1921)
the statistic has an approximate normal distribution with mean and variance
where .
For the transformed , the approximate variance is independent of the correlation . Furthermore, even the distribution of is not strictly normal, it tends to normality rapidly as the sample size increases for any values of (Fisher 1973, pp. 200–201).
For the null hypothesis , the p-values are computed by treating
as a normal random variable with mean zero and variance , where (Fisher 1973, p. 207; Anderson 1984, p. 123).
Note that the bias adjustment, , is always used when computing p-values under the null hypothesis in the CORR procedure.
The ALPHA= option in the FISHER option specifies the value for the confidence level , the RHO0= option specifies the value in the hypothesis , and the BIASADJ= option specifies whether the bias adjustment is to be used for the confidence limits.
The TYPE= option specifies the type of confidence limits. The TYPE=TWOSIDED option requests two-sided confidence limits and a p-value under the hypothesis . For a one-sided confidence limit, the TYPE=LOWER option requests a lower confidence limit and a p-value under the hypothesis , and the TYPE=UPPER option requests an upper confidence limit and a p-value under the hypothesis .
The confidence limits for the correlation are derived through the confidence limits for the parameter , with or without the bias adjustment.
Without a bias adjustment, confidence limits for are computed by treating
as having a normal distribution with mean zero and variance .
That is, the two-sided confidence limits for are computed as
where is the percentage point of the standard normal distribution.
With a bias adjustment, confidence limits for are computed by treating
as having a normal distribution with mean zero and variance , where the bias adjustment function (Keeping 1962, p. 308) is
That is, the two-sided confidence limits for are computed as
These computed confidence limits of and are then transformed back to derive the confidence limits for the correlation :
Note that with a bias adjustment, the CORR procedure also displays the following correlation estimate:
Fisher (1973, p. 199) describes the following practical applications of the z transformation:
testing whether a population correlation is equal to a given value
testing for equality of two population correlations
combining correlation estimates from different samples
To test if a population correlation from a sample of observations with sample correlation is equal to a given , first apply the z transformation to and : and .
The p-value is then computed by treating
as a normal random variable with mean zero and variance .
Assume that sample correlations and are computed from two independent samples of and observations, respectively. To test whether the two corresponding population correlations, and , are equal, first apply the z transformation to the two sample correlations: and .
The p-value is derived under the null hypothesis of equal correlation. That is, the difference is distributed as a normal random variable with mean zero and variance .
Assuming further that the two samples are from populations with identical correlation, a combined correlation estimate can be computed. The weighted average of the corresponding z values is
where the weights are inversely proportional to their variances.
Thus, a combined correlation estimate is and . See Example 2.4 for further illustrations of these applications.
Note that this approach can be extended to include more than two samples.