The CORR Procedure
- Overview
- Getting Started
-
Syntax
-
Details
Pearson Product-Moment Correlation Spearman Rank-Order Correlation Kendall’s Tau-b Correlation Coefficient Hoeffding Dependence Coefficient Partial Correlation Fisher’s z Transformation Polyserial Correlation Cronbach’s Coefficient Alpha Confidence and Prediction Ellipses Missing Values In-Database Computation Output Tables Output Data Sets ODS Table Names ODS Graphics
-
Examples
Computing Four Measures of Association Computing Correlations between Two Sets of Variables Analysis Using Fisher’s z Transformation Applications of Fisher’s z Transformation Computing Polyserial Correlations Computing Cronbach’s Coefficient Alpha Saving Correlations in an Output Data Set Creating Scatter Plots Computing Partial Correlations
- References
| Fisher’s z Transformation |
For a sample correlation 
that uses a sample from a bivariate normal distribution with correlation 
, the statistic
 |
has a Student’s 
distribution with (
) degrees of freedom.
With the monotone transformation of the correlation (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 1970, pp. 200–201).
For the null hypothesis 
, the 
-values are computed by treating
 |
as a normal random variable with mean zero and variance 
, where 
(Fisher 1970, p. 207; Anderson 1984, p. 123).
Note that the bias adjustment, 
, is always used when computing -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 -value under the hypothesis . For a one-sided confidence limit, the TYPE=LOWER option requests a lower confidence limit and a -value under the hypothesis 
, and the TYPE=UPPER option requests an upper confidence limit and a -value under the hypothesis 
.
Confidence Limits for the Correlation
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:
 |
Applications of Fisher’s z Transformation
Fisher (1970, p. 199) describes the following practical applications of the 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 transformation to and : 
and .
The -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 transformation to the two sample correlations: and 
.
The -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 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.