PROC CORR: Fisher’s z Transformation

The CORR Procedure

Fisher’s z Transformation

For a sample correlation $\text{[math]}$ that uses a sample from a bivariate normal distribution with correlation $\text{[math]}$ , the statistic

$\text{[math]}$

has a Student’s $\text{[math]}$ distribution with ( $\text{[math]}$ ) degrees of freedom.

With the monotone transformation of the correlation $\text{[math]}$ (Fisher 1921)

$\text{[math]}$

the statistic $\text{[math]}$ has an approximate normal distribution with mean and variance

$\text{[math]}$

where $\text{[math]}$ .

For the transformed $\text{[math]}$ , the approximate variance $\text{[math]}$ is independent of the correlation $\text{[math]}$ . Furthermore, even the distribution of $\text{[math]}$ is not strictly normal, it tends to normality rapidly as the sample size increases for any values of $\text{[math]}$ (Fisher 1970, pp. 200–201).

For the null hypothesis $\text{[math]}$ , the $\text{[math]}$ -values are computed by treating

$\text{[math]}$

as a normal random variable with mean zero and variance $\text{[math]}$ , where $\text{[math]}$ (Fisher 1970, p. 207; Anderson 1984, p. 123).

Note that the bias adjustment, $\text{[math]}$ , is always used when computing $\text{[math]}$ -values under the null hypothesis $\text{[math]}$ in the CORR procedure.

The ALPHA= option in the FISHER option specifies the value $\text{[math]}$ for the confidence level $\text{[math]}$ , the RHO0= option specifies the value $\text{[math]}$ in the hypothesis $\text{[math]}$ , 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 $\text{[math]}$ -value under the hypothesis $\text{[math]}$ . For a one-sided confidence limit, the TYPE=LOWER option requests a lower confidence limit and a $\text{[math]}$ -value under the hypothesis $\text{[math]}$ , and the TYPE=UPPER option requests an upper confidence limit and a $\text{[math]}$ -value under the hypothesis $\text{[math]}$ .

Confidence Limits for the Correlation

The confidence limits for the correlation $\text{[math]}$ are derived through the confidence limits for the parameter $\text{[math]}$ , with or without the bias adjustment.

Without a bias adjustment, confidence limits for $\text{[math]}$ are computed by treating

$\text{[math]}$

as having a normal distribution with mean zero and variance $\text{[math]}$ .

That is, the two-sided confidence limits for $\text{[math]}$ are computed as

$\text{[math]}$

where $\text{[math]}$ is the $\text{[math]}$ percentage point of the standard normal distribution.

With a bias adjustment, confidence limits for $\text{[math]}$ are computed by treating

$\text{[math]}$

as having a normal distribution with mean zero and variance $\text{[math]}$ , where the bias adjustment function (Keeping 1962, p. 308) is

$\text{[math]}$

That is, the two-sided confidence limits for $\text{[math]}$ are computed as

$\text{[math]}$

These computed confidence limits of $\text{[math]}$ and $\text{[math]}$ are then transformed back to derive the confidence limits for the correlation $\text{[math]}$ :

$\text{[math]}$

Note that with a bias adjustment, the CORR procedure also displays the following correlation estimate:

$\text{[math]}$

Applications of Fisher’s z Transformation

Fisher (1970, p. 199) describes the following practical applications of the $\text{[math]}$ 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 $\text{[math]}$ from a sample of $\text{[math]}$ observations with sample correlation $\text{[math]}$ is equal to a given $\text{[math]}$ , first apply the $\text{[math]}$ transformation to $\text{[math]}$ and $\text{[math]}$ : $\text{[math]}$ and $\text{[math]}$ .

The $\text{[math]}$ -value is then computed by treating

$\text{[math]}$

as a normal random variable with mean zero and variance $\text{[math]}$ .

Assume that sample correlations $\text{[math]}$ and $\text{[math]}$ are computed from two independent samples of $\text{[math]}$ and $\text{[math]}$ observations, respectively. To test whether the two corresponding population correlations, $\text{[math]}$ and $\text{[math]}$ , are equal, first apply the $\text{[math]}$ transformation to the two sample correlations: $\text{[math]}$ and $\text{[math]}$ .

The $\text{[math]}$ -value is derived under the null hypothesis of equal correlation. That is, the difference $\text{[math]}$ is distributed as a normal random variable with mean zero and variance $\text{[math]}$ .

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 $\text{[math]}$ values is

$\text{[math]}$

where the weights are inversely proportional to their variances.

Thus, a combined correlation estimate is $\text{[math]}$ and $\text{[math]}$ . See Example 2.4 for further illustrations of these applications.

Note that this approach can be extended to include more than two samples.

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